![](\"https:\/\/r2.wuhanqing.cn\/MyWebsiteFiles\/1-%E6%96%87%E7%AB%A0\/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0\/%E4%BB%8E%E7%82%B9%E7%A7%AF%E5%88%B0%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4%EF%BC%9A%E8%97%8F%E5%9C%A8%E5%BE%AE%E7%A7%AF%E5%88%86%E3%80%81%E4%BF%A1%E5%8F%B7%E4%B8%8EAI%E8%83%8C%E5%90%8E%E7%9A%84%E5%90%8C%E4%B8%80%E5%A5%97%E8%AF%AD%E8%A8%80\/Pictures\/01_cosine_similarity_heatmap.png\")
<\/p>\n# \u30c9\u30c3\u30c8\u7a4d\u304b\u3089\u5185\u7a4d\u7a7a\u9593\u3078\uff1a\u7dda\u5f62\u4ee3\u6570\u3001\u4fe1\u53f7\u51e6\u7406\u3001AI\u306e\u80cc\u5f8c\u306b\u3042\u308b\u7d71\u4e00\u3055\u308c\u305f\u8a00\u8a9e (From Dot Product to Inner Product Space: The Unified Language Behind Linear Algebra, Signals, and AI)<\/p>\n
## \u8981\u7d04 (Abstract)<\/p>\n
**\u5185\u7a4d(Inner Product)** \u306f\u3001\u7dda\u5f62\u4ee3\u6570\u3001\u95a2\u6570\u89e3\u6790\u5b66\u3001\u4fe1\u53f7\u51e6\u7406\u3001\u6a5f\u68b0\u5b66\u7fd2\u3001\u91cf\u5b50\u529b\u5b66\u306b\u308f\u305f\u3063\u3066\u5171\u6709\u3055\u308c\u308b\u6838\u5fc3\u7684\u306a\u4ee3\u6570\u69cb\u9020\u3067\u3042\u308b\u3002\u672c\u8ad6\u6587\u306f\u300c\u5185\u7a4d\u300d\u3092\u552f\u4e00\u306e\u30c6\u30fc\u30de\u3068\u3057\u3001\u6709\u9650\u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u304a\u3051\u308b\u30c9\u30c3\u30c8\u7a4d(Dot Product)\u304b\u3089\u51fa\u767a\u3057\u3001\u5185\u7a4d\u7a7a\u9593\u306e\u516c\u7406\u3001\u76f4\u4ea4\u5206\u89e3(Orthogonal Decomposition)\u3001\u6700\u5c0f\u4e8c\u4e57\u5c04\u5f71(Least-Squares Projection)\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593(Hilbert Space)\u3001\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3068\u5909\u63db(Fourier Series and Transform)\u3001\u7573\u307f\u8fbc\u307f(Convolution)\u3001\u96e2\u6563\u30b3\u30b5\u30a4\u30f3\u5909\u63db(Discrete Cosine Transform)\u3001\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u89e3\u6790(Wavelet Analysis)\u3001\u81ea\u5df1\u6ce8\u610f\u30e1\u30ab\u30cb\u30ba\u30e0(Self-Attention Mechanism)\u3001\u30ab\u30fc\u30cd\u30eb\u6cd5(Kernel Method)\u3001\u305d\u3057\u3066\u91cf\u5b50\u529b\u5b66\u306b\u304a\u3051\u308b\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u5c04\u5f71(State-Vector Projection)\u3092\u9806\u306b\u7d39\u4ecb\u3059\u308b\u3002\u4e00\u898b\u7570\u306a\u308b\u5b66\u554f\u5206\u91ce\u306b\u5c5e\u3059\u308b\u3088\u3046\u306b\u898b\u3048\u308b\u3053\u308c\u3089\u306e\u6982\u5ff5\u304c\u3001\u6570\u5b66\u7684\u69cb\u9020\u306b\u304a\u3044\u3066\u7d71\u4e00\u6027\u3092\u6301\u3064\u3053\u3068\u3092\u660e\u3089\u304b\u306b\u3059\u308b\uff1a**\u5185\u7a4d\u306e\u5b9a\u7fa9 \u2192 \u76f4\u4ea4\u57fa\u5e95\u306e\u78ba\u7acb \u2192 \u5c04\u5f71\u5206\u89e3 \u2192 \u60c5\u5831\u62bd\u51fa**\u3002\u672c\u8ad6\u6587\u306f\u8aad\u8005\u306b\u6570\u5b66\u3001\u5de5\u5b66\u3001\u7269\u7406\u5b66\u3092\u8cab\u304f\u8a8d\u77e5\u5730\u56f3(Cognitive Map)\u3092\u63d0\u4f9b\u3059\u308b\u3053\u3068\u3092\u76ee\u7684\u3068\u3059\u308b\u3002<\/p>\n
## \u5e8f\u6587\uff1a\u4e07\u7269\u306f\u5c04\u5f71\u3067\u3042\u308b (Preface: Everything Is a Projection)<\/p>\n
\u6570\u5b66\u3068\u5de5\u5b66\u79d1\u5b66\u306b\u306f\u7e70\u308a\u8fd4\u3057\u73fe\u308c\u308b\u30d1\u30bf\u30fc\u30f3\u304c\u3042\u308b\uff1a\u8907\u96d1\u306a\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3092\u8907\u6570\u306e\u300c\u57fa\u672c\u6210\u5206\u300d\u306e\u7dda\u5f62\u7d50\u5408\u306b\u5206\u89e3\u3059\u308b\u3053\u3068\u3001\u305d\u3057\u3066\u5206\u89e3\u306e\u9053\u5177\u304c\u307e\u3055\u306b**\u5c04\u5f71(Projection)** \u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3060\u3002\u5c04\u5f71\u6f14\u7b97\u306e\u672c\u8cea\u306f\u5185\u7a4d(Inner Product)\u3067\u3042\u308b\u2014\u300c\u985e\u4f3c\u6027(Similarity)\u300d\u3092\u6e2c\u5b9a\u3059\u308b\u4e8c\u9805\u6f14\u7b97\u3002\u30d5\u30fc\u30ea\u30a8\u89e3\u6790\u3067\u4fe1\u53f7\u3092\u7570\u306a\u308b\u5468\u6ce2\u6570\u306e\u6b63\u5f26\u6ce2\u306b\u5206\u89e3\u3059\u308b\u3053\u3068\u304b\u3089\u3001\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3067\u30c7\u30fc\u30bf\u306b\u6700\u3082\u3088\u304f\u9069\u5408\u3059\u308b\u76f4\u7dda\u3092\u898b\u3064\u3051\u308b\u3053\u3068\u3001\u91cf\u5b50\u529b\u5b66\u3067\u91cd\u306d\u5408\u308f\u305b\u72b6\u614b\u306b\u3042\u308b\u7c92\u5b50\u3092\u6e2c\u5b9a\u3059\u308b\u3053\u3068\u307e\u3067\u3001\u3053\u308c\u3089\u3059\u3079\u3066\u306e\u30d7\u30ed\u30bb\u30b9\u306f\u540c\u4e00\u306e\u6570\u5b66\u8a00\u8a9e\u3092\u5171\u6709\u3057\u3066\u3044\u308b\uff1a**\u5185\u7a4d\u306e\u5b9a\u7fa9 \u2192 \u76f4\u4ea4\u57fa\u5e95\u306e\u78ba\u7acb \u2192 \u5c04\u5f71 \u2192 \u76f4\u4ea4\u5206\u89e3 \u2192 \u60c5\u5831\u62bd\u51fa**\u3002<\/p>\n
\u672c\u8ad6\u6587\u306e\u76ee\u6a19\u306f\u3001\u3053\u306e\u7d71\u4e00\u3055\u308c\u305f\u30d5\u30ec\u30fc\u30e0\u30ef\u30fc\u30af\u3092\u4f53\u7cfb\u7684\u306b\u8aac\u660e\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u6700\u3082\u8eab\u8fd1\u306a\u30d9\u30af\u30c8\u30eb\u306e\u30c9\u30c3\u30c8\u7a4d(Dot Product)\u304b\u3089\u51fa\u767a\u3057\u3001\u5f90\u3005\u306b\u5185\u7a4d\u7a7a\u9593(Inner Product Space)\u3068\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593(Hilbert Space)\u3078\u3068\u62bd\u8c61\u5316\u3057\u3001\u3053\u306e\u69cb\u9020\u304c\u5fae\u7a4d\u5206\u5b66\u3001\u4fe1\u53f7\u51e6\u7406\u3001\u4eba\u5de5\u77e5\u80fd\u3001\u91cf\u5b50\u529b\u5b66\u306b\u304a\u3044\u3066\u3069\u306e\u3088\u3046\u306b\u7e70\u308a\u8fd4\u3057\u73fe\u308c\u308b\u304b\u3092\u793a\u3059\u3002\u8aad\u8005\u306b\u95a2\u6570\u89e3\u6790\u5b66(Functional Analysis)\u306e\u80cc\u666f\u77e5\u8b58\u306f\u5fc5\u8981\u306a\u304f\u3001\u57fa\u672c\u7684\u306a\u7dda\u5f62\u4ee3\u6570\u3068\u5fae\u7a4d\u5206\u5b66\u306e\u77e5\u8b58\u304c\u3042\u308c\u3070\u5341\u5206\u3067\u3042\u308b\u3002<\/p>\n
---<\/p>\n
## \u7b2c1\u7ae0 \u5185\u7a4d\u306e\u672c\u4f53 \u2014 \u985e\u4f3c\u6027\u3092\u6e2c\u5b9a\u3059\u308b\u57fa\u672c\u6f14\u7b97 (Chapter 1 The Ontology of Inner Products \u2014 The Fundamental Operation for Measuring Similarity)<\/p>\n
### 1.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n
\u5185\u7a4d(Inner Product)\u306e\u6982\u5ff5\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u5e7e\u4f55\u5b66\u306e\u30c9\u30c3\u30c8\u7a4d(Dot Product)\u306b\u8d77\u6e90\u3092\u6301\u3064\u304c\u3001\u305d\u306e\u6570\u5b66\u7684\u610f\u5473\u306f\u95a2\u6570\u89e3\u6790\u5b66(Functional Analysis)\u306b\u304a\u3044\u3066\u5927\u304d\u304f\u62e1\u5f35\u3055\u308c\u305f\u3002\u672c\u7bc0\u306f\u6709\u9650\u6b21\u5143\u306e\u5834\u5408\u304b\u3089\u51fa\u767a\u3057\u3001\u5f90\u3005\u306b\u5185\u7a4d\u306e\u53b3\u5bc6\u306a\u5b9a\u7fa9\u3092\u69cb\u7bc9\u3059\u308b\u3002<\/p>\n
```ad-definition
\ntitle: \u5b9a\u7fa9 1.1 \u30c9\u30c3\u30c8\u7a4d (Definition 1.1 Dot Product)
\n$\\mathbb{R}^n$\u3092$n$\u6b21\u5143\u5b9f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u3068\u3059\u308b\u3002\u4efb\u610f\u306e\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\mathbf{a} = (a_1, a_2, \\dots, a_n)$ \u3068 $\\mathbf{b} = (b_1, b_2, \\dots, b_n)$ \u306b\u3064\u3044\u3066\u3001\u305d\u306e\u30c9\u30c3\u30c8\u7a4d\u306f\u5bfe\u5fdc\u3059\u308b\u6210\u5206\u306e\u7a4d\u306e\u548c\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b$^{[1]}$:<\/p>\n
$$
\n\\langle \\mathbf{a}, \\mathbf{b} \\rangle = \\mathbf{a} \\cdot \\mathbf{b} = \\sum_{i=1}^{n} a_i b_i.
\n\\tag{1.1}
\n$$<\/p>\n
\u30c9\u30c3\u30c8\u7a4d\u306f\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u3092\u4e00\u3064\u306e\u30b9\u30ab\u30e9\u30fc\u306b\u5199\u50cf\u3059\u308b\u4e8c\u9805\u6f14\u7b97\u3067\u3042\u308b\u3002\u305d\u306e\u5e7e\u4f55\u5b66\u7684\u89e3\u91c8\u306f\u30b3\u30b5\u30a4\u30f3\u6cd5\u5247(Cosine Law)\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b:<\/p>\n
$$
\n\\mathbf{a} \\cdot \\mathbf{b} = \\|\\mathbf{a}\\| \\|\\mathbf{b}\\| \\cos\\theta,
\n\\tag{1.2}
\n$$<\/p>\n
\u3053\u3053\u3067 $\\|\\mathbf{a}\\| = \\sqrt{\\langle \\mathbf{a}, \\mathbf{a} \\rangle}$ \u306f\u30d9\u30af\u30c8\u30eb\u306e\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u30ce\u30eb\u30e0($L_2$ \u30ce\u30eb\u30e0)\u3067\u3042\u308a\u3001$\\theta$ \u306f\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u9593\u306e\u89d2\u5ea6\u3067\u3042\u308b\u3002
\n```<\/p>\n
```ad-definition
\ntitle: \u5b9a\u7fa9 1.2 \u5185\u7a4d\u7a7a\u9593 (Definition 1.2 Inner Product Space)
\n$V$ \u3092\u4f53 $\\mathbb{F}$($\\mathbb{R}$ \u307e\u305f\u306f $\\mathbb{C}$) \u4e0a\u306e\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u3068\u3059\u308b\u3002\u5199\u50cf $\\langle \\cdot, \\cdot \\rangle: V \\times V \\to \\mathbb{F}$ \u304c\u6b21\u306e\u4e09\u3064\u306e\u516c\u7406\u3092\u6e80\u305f\u3059\u3068\u304d\u3001\u5185\u7a4d(Inner Product)\u3068\u3044\u3046$^{[8][9]}$:<\/p>\n
1. **\u5171\u5f79\u5bfe\u79f0\u6027(Conjugate Symmetry)**: $\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\overline{\\langle \\mathbf{v}, \\mathbf{u} \\rangle}$\u3001\u3053\u3053\u3067\u4e0a\u7dda\u306f\u8907\u7d20\u5171\u5f79\u3092\u8868\u3059\u3002\u5b9f\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u3067\u306f\u5bfe\u79f0\u6027 $\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\langle \\mathbf{v}, \\mathbf{u} \\rangle$ \u306b\u7e2e\u6e1b\u3055\u308c\u308b\u3002
\n2. **\u7b2c\u4e00\u5909\u6570\u306b\u95a2\u3059\u308b\u7dda\u5f62\u6027(Linearity in the First Argument)**: $\\langle \\alpha\\mathbf{u} + \\beta\\mathbf{v}, \\mathbf{w} \\rangle = \\alpha\\langle \\mathbf{u}, \\mathbf{w} \\rangle + \\beta\\langle \\mathbf{v}, \\mathbf{w} \\rangle$\u3001\u4efb\u610f\u306e $\\alpha, \\beta \\in \\mathbb{F}$ \u306b\u3064\u3044\u3066\u6210\u7acb\u3002
\n3. **\u6b63\u5b9a\u5024\u6027(Positive Definiteness)**: $\\langle \\mathbf{v}, \\mathbf{v} \\rangle \\geq 0$\u3001\u304b\u3064 $\\langle \\mathbf{v}, \\mathbf{v} \\rangle = 0$ \u3068\u306a\u308b\u306e\u306f $\\mathbf{v} = \\mathbf{0}$ \u306e\u5834\u5408\u306e\u307f\u3002<\/p>\n
\u5185\u7a4d\u304b\u3089\u30ce\u30eb\u30e0 $\\|\\mathbf{v}\\| = \\sqrt{\\langle \\mathbf{v}, \\mathbf{v} \\rangle}$ \u304c\u5c0e\u304b\u308c\u3001\u305d\u3053\u304b\u3089\u8ddd\u96e2 $d(\\mathbf{u}, \\mathbf{v}) = \\|\\mathbf{u} - \\mathbf{v}\\|$ \u304c\u5c0e\u304b\u308c\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u5185\u7a4d\u7a7a\u9593\u306f\u81ea\u7136\u306b\u30ce\u30eb\u30e0\u7a7a\u9593(Normed Space)\u3068\u306a\u308a\u3001\u3055\u3089\u306b\u8ddd\u96e2\u7a7a\u9593(Metric Space)\u3068\u306a\u308b\u3002
\n```<\/p>\n
```ad-theorem
\ntitle: \u5b9a\u7406 1.1 \u30b3\u30fc\u30b7\u30fc-\u30b7\u30e5\u30ef\u30eb\u30c4\u4e0d\u7b49\u5f0f (Theorem 1.1 Cauchy-Schwarz Inequality)
\n\u5185\u7a4d\u7a7a\u9593 $V$ \u306e\u4efb\u610f\u306e\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\mathbf{u}, \\mathbf{v}$ \u306b\u3064\u3044\u3066\u6b21\u304c\u6210\u7acb\u3059\u308b$^{[8]}$:<\/p>\n
$$
\n|\\langle \\mathbf{u}, \\mathbf{v} \\rangle| \\leq \\|\\mathbf{u}\\| \\cdot \\|\\mathbf{v}\\|.
\n\\tag{1.3}
\n$$<\/p>\n
\u7b49\u53f7\u306f $\\mathbf{u}$ \u3068 $\\mathbf{v}$ \u304c\u7dda\u5f62\u5f93\u5c5e\u3067\u3042\u308b\u3068\u304d\uff08\u3059\u306a\u308f\u3061\u3001\u4e00\u65b9\u304c\u4ed6\u65b9\u306e\u30b9\u30ab\u30e9\u30fc\u500d\u3067\u3042\u308b\u3068\u304d\uff09\u306b\u6210\u7acb\u3059\u308b\u3002
\n```<\/p>\n
```ad-definition
\ntitle: \u5b9a\u7fa9 1.3 \u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6 (Definition 1.3 Cosine Similarity)
\n\u4e8c\u3064\u306e\u975e\u96f6\u30d9\u30af\u30c8\u30eb $\\mathbf{a}, \\mathbf{b} \\in \\mathbb{R}^n$ \u306b\u3064\u3044\u3066\u3001\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u306f\u6b63\u898f\u5316\u3055\u308c\u305f\u5185\u7a4d\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b$^{[13]}$:<\/p>\n
$$
\n\\text{cosine\\_similarity}(\\mathbf{a}, \\mathbf{b}) = \\frac{\\langle \\mathbf{a}, \\mathbf{b} \\rangle}{\\|\\mathbf{a}\\| \\|\\mathbf{b}\\|} = \\cos\\theta.
\n\\tag{1.5}
\n$$<\/p>\n
\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3092\u9664\u53bb\u3057\u65b9\u5411\u306e\u985e\u4f3c\u6027\u306e\u307f\u3092\u6e2c\u5b9a\u3059\u308b\u305f\u3081\u3001\u6587\u66f8\u5206\u985e\u3001\u610f\u5473\u691c\u7d22\u306a\u3069\u3067\u5e83\u304f\u4f7f\u7528\u3055\u308c\u308b\u3002
\n```<\/p>\n
### 1.2 \u5e7e\u4f55\u5b66\u3068\u7a7a\u9593\u7684\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)<\/p>\n
\u5185\u7a4d\u306e\u5e7e\u4f55\u5b66\u7684\u610f\u5473\u306f**\u5c04\u5f71(Projection)** \u3067\u3042\u308b\u3002$\\langle \\mathbf{a}, \\mathbf{b} \\rangle$ \u306f\u30d9\u30af\u30c8\u30eb $\\mathbf{a}$ \u3092 $\\mathbf{b}$ \u65b9\u5411\u306b\u5c04\u5f71\u3057\u305f\u9577\u3055\u306b $\\|\\mathbf{b}\\|$ \u3092\u4e57\u3058\u305f\u5024\u3067\u3042\u308b\u3002$\\mathbf{b}$ \u304c\u5358\u4f4d\u30d9\u30af\u30c8\u30eb(Unit Vector)\u306e\u3068\u304d\u3001\u5185\u7a4d\u306f\u6b63\u78ba\u306b $\\mathbf{a}$ \u306e $\\mathbf{b}$ \u4e0a\u3078\u306e\u5c04\u5f71\u9577\u3068\u306a\u308b\u3002<\/p>\n
\u3053\u306e\u89b3\u70b9\u304b\u3089\u3001\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u306f\u4e8c\u3064\u306e\u30d9\u30af\u30c8\u30eb\u65b9\u5411\u306e\u6574\u5217\u5ea6\u3092\u6e2c\u5b9a\u3059\u308b:
\n- $\\cos\\theta = 1$: \u540c\u4e00\u65b9\u5411\u3001\u6700\u5927\u985e\u4f3c\u5ea6;
\n- $\\cos\\theta = 0$: \u76f4\u4ea4\u3001\u985e\u4f3c\u5ea6 0;
\n- $\\cos\\theta = -1$: \u53cd\u5bfe\u65b9\u5411\u3001\u6700\u5927\u53cd\u985e\u4f3c\u5ea6\u3002<\/p>\n
### 1.3 \u30cf\u30fc\u30c9\u30b3\u30a2\u4f8b\u984c\u8a73\u7d30\u89e3\u8aac (Worked Example)<\/p>\n
```ad-example
\ntitle: \u4f8b\u984c 1.1 \u30b0\u30e9\u30e0-\u30b7\u30e5\u30df\u30c3\u30c8\u76f4\u4ea4\u5316 (Example 1.1 Gram-Schmidt Orthogonalization)
\n$\\mathbb{R}^3$ \u306b\u304a\u3044\u3066\u4e09\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\mathbf{v}_1 = (1, 1, 0)$, $\\mathbf{v}_2 = (1, 0, 1)$, $\\mathbf{v}_3 = (0, 1, 1)$ \u304c\u4e0e\u3048\u3089\u308c\u305f\u3002\u3053\u308c\u3089\u306e\u30d9\u30af\u30c8\u30eb\u304b\u3089\u30b0\u30e9\u30e0-\u30b7\u30e5\u30df\u30c3\u30c8\u904e\u7a0b\u3092\u7528\u3044\u3066\u4e00\u7d44\u306e\u76f4\u4ea4\u57fa\u5e95\u3092\u69cb\u6210\u305b\u3088\u3002<\/p>\n
**\u89e3**:<\/p>\n
**\u30b9\u30c6\u30c3\u30d7 1**: $\\mathbf{u}_1 = \\mathbf{v}_1 = (1, 1, 0)$ \u3068\u8a2d\u5b9a\u3002$\\|\\mathbf{u}_1\\| = \\sqrt{2}$\u3002<\/p>\n
**\u30b9\u30c6\u30c3\u30d7 2**: $\\mathbf{v}_2$ \u304b\u3089 $\\mathbf{u}_1$ \u4e0a\u3078\u306e\u5c04\u5f71\u3092\u9664\u53bb:<\/p>\n
$$
\n\\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_2) = \\frac{\\langle \\mathbf{v}_2, \\mathbf{u}_1 \\rangle}{\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle} \\mathbf{u}_1 = \\frac{1}{2} (1, 1, 0) = (0.5, 0.5, 0)
\n$$<\/p>\n
$$
\n\\mathbf{u}_2 = \\mathbf{v}_2 - \\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_2) = (1, 0, 1) - (0.5, 0.5, 0) = (0.5, -0.5, 1)
\n$$<\/p>\n
**\u30b9\u30c6\u30c3\u30d7 3**: $\\mathbf{v}_3$ \u304b\u3089 $\\mathbf{u}_1$ \u3068 $\\mathbf{u}_2$ \u4e0a\u3078\u306e\u5c04\u5f71\u3092\u9664\u53bb:<\/p>\n
$$
\n\\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_3) = \\frac{0}{2} \\mathbf{u}_1 = (0, 0, 0), \\quad
\n\\text{proj}_{\\mathbf{u}_2}(\\mathbf{v}_3) = \\frac{\\langle \\mathbf{v}_3, \\mathbf{u}_2 \\rangle}{\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle} \\mathbf{u}_2
\n$$<\/p>\n
$\\langle \\mathbf{v}_3, \\mathbf{u}_2 \\rangle = 0 \\times 0.5 + 1 \\times (-0.5) + 1 \\times 1 = 0.5$, $\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle = 0.25 + 0.25 + 1 = 1.5$.<\/p>\n
$$
\n\\text{proj}_{\\mathbf{u}_2}(\\mathbf{v}_3) = \\frac{0.5}{1.5} (0.5, -0.5, 1) = \\left(\\frac{1}{6}, -\\frac{1}{6}, \\frac{1}{3}\\right)
\n$$<\/p>\n
$$
\n\\mathbf{u}_3 = \\mathbf{v}_3 - \\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_3) - \\text{proj}_{\\mathbf{u}_2}(\\mathbf{v}_3) = \\left(-\\frac{1}{6}, \\frac{7}{6}, \\frac{2}{3}\\right)
\n$$<\/p>\n
**\u691c\u8a3c**: $\\langle \\mathbf{u}_1, \\mathbf{u}_2 \\rangle = 0.5 - 0.5 + 0 = 0$\u3002\u76f4\u4ea4\u6027\u304c\u78ba\u8a8d\u3055\u308c\u305f\u3002
\n```<\/p>\n
### 1.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)<\/p>\n
\u5185\u7a4d\u3068\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u306f\u81ea\u7136\u8a00\u8a9e\u51e6\u7406(NLP)\u306b\u304a\u3044\u3066\u5358\u8a9e\u57cb\u3081\u8fbc\u307f(Word Embedding)\u306e\u610f\u5473\u7684\u985e\u4f3c\u5ea6\u3092\u6e2c\u5b9a\u3059\u308b\u6a19\u6e96\u30c4\u30fc\u30eb\u3067\u3042\u308b$^{[21]}$\u3002Word2Vec\u3001GloVe\u306a\u3069\u306e\u57cb\u3081\u8fbc\u307f\u30e2\u30c7\u30eb\u306f\u5404\u5358\u8a9e\u3092\u9ad8\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306b\u5199\u50cf\u3057\u3001\u5358\u8a9e\u9593\u306e\u610f\u5473\u7684\u985e\u4f3c\u5ea6\u306f\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u3067\u5b9a\u91cf\u5316\u3055\u308c\u308b\u3002<\/p>\n
\u56f31\u306f5\u3064\u306e\u5358\u8a9e\u57cb\u3081\u8fbc\u307f\u9593\u306e\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u30d2\u30fc\u30c8\u30de\u30c3\u30d7\u3092\u793a\u3059\u3002\"king-queen\"\u3001\"man-woman\"\u306e\u30da\u30a2\u306f\u985e\u4f3c\u5ea6\u304c\u9ad8\u304f(\u660e\u308b\u3044\u8272)\u3001\"apple\"\u306f\u4ed6\u306e\u5358\u8a9e\u3068\u306e\u985e\u4f3c\u5ea6\u304c\u4f4e\u3044(\u6697\u3044\u8272)\u3002\u3053\u308c\u306f\u5185\u7a4d\u304c\u5358\u306a\u308b\u6570\u5b66\u6f14\u7b97\u3092\u8d85\u3048\u3066\u610f\u5473\u7684\u95a2\u4fc2\u3092\u6355\u6349\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n
<\/p>\n
**\u56f31: \u5358\u8a9e\u57cb\u3081\u8fbc\u307f\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u30d2\u30fc\u30c8\u30de\u30c3\u30d7(Figure 1: Cosine Similarity Heatmap of Word Embeddings).** \"king-queen\"\u3068\"man-woman\"\u306f\u610f\u5473\u7684\u985e\u4f3c\u6027\u304c\u9ad8\u304f\u3001\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u304c\u9ad8\u3044\u3002\"apple\"\u306f\u679c\u7269\u30ab\u30c6\u30b4\u30ea\u306b\u5c5e\u3059\u308b\u305f\u3081\u4ed6\u306e\u5358\u8a9e\u3068\u306e\u985e\u4f3c\u5ea6\u304c\u4f4e\u3044\u3002<\/p>\n
---<\/p>\n
## \u7b2c2\u7ae0 \u76f4\u4ea4\u5206\u89e3 \u2014 \u8907\u96d1\u306a\u3082\u306e\u3092\u5206\u96e2\u3059\u308b\u6280\u8853 (Chapter 2 Orthogonal Decomposition \u2014 The Art of Separating Complex Things)<\/p>\n
### 2.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n
\u76f4\u4ea4\u5206\u89e3(Orthogonal Decomposition)\u306f\u5185\u7a4d\u7a7a\u9593\u306b\u304a\u3044\u3066\u6700\u3082\u5f37\u529b\u306a\u30c4\u30fc\u30eb\u306e\u4e00\u3064\u3067\u3042\u308b\u3002\u305d\u306e\u6838\u3068\u306a\u308b\u30a2\u30a4\u30c7\u30a2\u306f\uff1a\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u3092\u76f8\u4e92\u306b\u76f4\u4ea4\u3059\u308b\u90e8\u5206\u7a7a\u9593\u306e\u76f4\u548c(Direct Sum)\u306b\u5206\u89e3\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n
```ad-definition
\ntitle: \u5b9a\u7fa9 2.1 \u76f4\u4ea4\u88dc\u7a7a\u9593 (Definition 2.1 Orthogonal Complement)
\n$V$ \u3092\u5185\u7a4d\u7a7a\u9593\u3068\u3057\u3001$W \\subseteq V$ \u3092\u90e8\u5206\u7a7a\u9593\u3068\u3059\u308b\u3002$W$ \u306e\u76f4\u4ea4\u88dc\u7a7a\u9593(Orthogonal Complement) $W^\\perp$ \u306f $W$ \u306e\u3059\u3079\u3066\u306e\u30d9\u30af\u30c8\u30eb\u3068\u76f4\u4ea4\u3059\u308b\u30d9\u30af\u30c8\u30eb\u306e\u96c6\u5408\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b$^{[2]}$:<\/p>\n
$$
\nW^\\perp = \\{ \\mathbf{v} \\in V \\mid \\langle \\mathbf{v}, \\mathbf{w} \\rangle = 0,\\ \\forall \\mathbf{w} \\in W \\}.
\n\\tag{2.1}
\n$$
\n```<\/p>\n
```ad-theorem
\ntitle: \u5b9a\u7406 2.1 \u76f4\u4ea4\u5206\u89e3\u5b9a\u7406 (Theorem 2.1 Orthogonal Decomposition Theorem)
\n$W$ \u3092\u5185\u7a4d\u7a7a\u9593 $V$ \u306e\u6709\u9650\u6b21\u5143\u90e8\u5206\u7a7a\u9593\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u4efb\u610f\u306e $\\mathbf{v} \\in V$ \u306f\u4e00\u610f\u306b\u6b21\u306e\u3088\u3046\u306b\u5206\u89e3\u3055\u308c\u308b$^{[2]}$:<\/p>\n
$$
\n\\mathbf{v} = \\mathbf{w} + \\mathbf{w}^\\perp,
\n\\tag{2.2}
\n$$<\/p>\n
\u3053\u3053\u3067 $\\mathbf{w} \\in W$ \u304b\u3064 $\\mathbf{w}^\\perp \\in W^\\perp$ \u3067\u3042\u308b\u3002\u3059\u306a\u308f\u3061 $V = W \\oplus W^\\perp$ \u3067\u3042\u308b\u3002$\\mathbf{w}$ \u3092 $W$ \u4e0a\u3078\u306e $\\mathbf{v}$ \u306e\u76f4\u4ea4\u5c04\u5f71(Orthogonal Projection)\u3068\u3044\u3044\u3001$\\text{proj}_W(\\mathbf{v})$ \u3068\u8868\u8a18\u3059\u308b\u3002
\n```<\/p>\n
```ad-theorem
\ntitle: \u5b9a\u7406 2.2 \u76f4\u4ea4\u57fa\u5e95\u4e0a\u3078\u306e\u5c04\u5f71 (Theorem 2.2 Projection onto an Orthogonal Basis)
\n$\\{\\mathbf{u}_1, \\dots, \\mathbf{u}_k\\}$ \u304c\u90e8\u5206\u7a7a\u9593 $W$ \u306e\u76f4\u4ea4\u57fa\u5e95(Orthogonal Basis)\u3067\u3042\u308b\u3068\u3059\u308b\u3002$W$ \u4e0a\u3078\u306e $\\mathbf{v}$ \u306e\u76f4\u4ea4\u5c04\u5f71\u306f\u3001\u5404\u57fa\u5e95\u65b9\u5411\u3078\u306e\u5c04\u5f71\u306e\u548c\u3067\u4e0e\u3048\u3089\u308c\u308b:<\/p>\n
$$
\n\\text{proj}_W(\\mathbf{v}) = \\sum_{i=1}^{k} \\frac{\\langle \\mathbf{v}, \\mathbf{u}_i \\rangle}{\\langle \\mathbf{u}_i, \\mathbf{u}_i \\rangle} \\mathbf{u}_i.
\n\\tag{2.3}
\n$$<\/p>\n
\u57fa\u5e95\u304c\u6b63\u898f\u76f4\u4ea4(Orthonormal)\u3067\u3042\u308c\u3070($\\|\\mathbf{u}_i\\| = 1$)\u3001\u516c\u5f0f\u306f\u3088\u308a\u7c21\u5358\u306b\u306a\u308b:<\/p>\n
$$
\n\\text{proj}_W(\\mathbf{v}) = \\sum_{i=1}^{k} \\langle \\mathbf{v}, \\mathbf{u}_i \\rangle \\mathbf{u}_i.
\n\\tag{2.4}
\n$$
\n```<\/p>\n
### 2.2 \u5e7e\u4f55\u5b66\u3068\u7a7a\u9593\u7684\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)<\/p>\n
\u76f4\u4ea4\u5206\u89e3\u306e\u5e7e\u4f55\u5b66\u7684\u610f\u5473\u306f\u975e\u5e38\u306b\u76f4\u611f\u7684\u3067\u3042\u308b\uff1a3\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3044\u3066\u30d9\u30af\u30c8\u30eb $\\mathbf{v}$ \u306f $xy$-\u5e73\u9762\u4e0a\u3078\u306e\u5c04\u5f71 $\\mathbf{w}$ \u3068 $z$ \u8ef8\u65b9\u5411\u306e\u6210\u5206 $\\mathbf{w}^\\perp$ \u306b\u5206\u89e3\u3067\u304d\u308b\u3002$\\mathbf{w}$ \u306f $\\mathbf{v}$ \u304b\u3089 $W$ \u306b\u5782\u76f4\u306a\u6210\u5206\u3092\u9664\u53bb\u3057\u3066\u5f97\u3089\u308c\u308b\u3002<\/p>\n
\u5f0f (2.3) \u306e\u5404\u9805 $\\frac{\\langle \\mathbf{v}, \\mathbf{u}_i \\rangle}{\\langle \\mathbf{u}_i, \\mathbf{u}_i \\rangle} \\mathbf{u}_i$ \u306f $\\mathbf{v}$ \u3092 $\\mathbf{u}_i$ \u65b9\u5411\u306b\u5c04\u5f71\u3057\u305f\u3082\u306e\u3067\u3042\u308b\u3002\u76f4\u4ea4\u57fa\u5e95\u306e\u5229\u70b9\u306f\u3001\u5404\u65b9\u5411\u3078\u306e\u5c04\u5f71\u304c\u4e92\u3044\u306b\u72ec\u7acb\u3067\u3042\u308b\u3053\u3068\u3067\u3042\u308b\u2014\u3042\u308b\u65b9\u5411\u306e\u5c04\u5f71\u3092\u5909\u66f4\u3057\u3066\u3082\u4ed6\u306e\u65b9\u5411\u306e\u5c04\u5f71\u306b\u5f71\u97ff\u3092\u4e0e\u3048\u306a\u3044\u3002<\/p>\n
### 2.3 \u30cf\u30fc\u30c9\u30b3\u30a2\u4f8b\u984c\u8a73\u7d30\u89e3\u8aac (Worked Example)<\/p>\n
```ad-example
\ntitle: \u4f8b\u984c 2.1 \u76f4\u4ea4\u57fa\u5e95\u4e0a\u3078\u306e\u5c04\u5f71 (Example 2.1 Projection onto an Orthogonal Basis)
\n$\\mathbb{R}^3$ \u306b\u304a\u3044\u3066\u90e8\u5206\u7a7a\u9593 $W = \\text{span}\\{\\mathbf{u}_1, \\mathbf{u}_2\\}$ \u3092\u8003\u3048\u308b\u3002\u3053\u3053\u3067 $\\mathbf{u}_1 = (1, 1, 0)$, $\\mathbf{u}_2 = (0, 0, 1)$ \u3067\u3042\u308b\u3002$\\mathbf{v} = (3, 1, 2)$ \u3092 $W$ \u4e0a\u306b\u5c04\u5f71\u305b\u3088\u3002<\/p>\n
**\u89e3**: \u307e\u305a $\\mathbf{u}_1$ \u3068 $\\mathbf{u}_2$ \u304c\u76f4\u4ea4\u3059\u308b\u3053\u3068\u3092\u78ba\u8a8d: $\\langle \\mathbf{u}_1, \\mathbf{u}_2 \\rangle = 1 \\times 0 + 1 \\times 0 + 0 \\times 1 = 0$\u3002\u76f4\u4ea4\u3057\u3066\u3044\u308b\u3002<\/p>\n
\u5f0f (2.3) \u3092\u4f7f\u7528:<\/p>\n
$$
\n\\text{proj}_W(\\mathbf{v}) = \\frac{\\langle \\mathbf{v}, \\mathbf{u}_1 \\rangle}{\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle} \\mathbf{u}_1 + \\frac{\\langle \\mathbf{v}, \\mathbf{u}_2 \\rangle}{\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle} \\mathbf{u}_2.
\n$$<\/p>\n
\u5404\u9805\u3092\u8a08\u7b97:<\/p>\n
$$
\n\\frac{\\langle \\mathbf{v}, \\mathbf{u}_1 \\rangle}{\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle} = \\frac{3 \\times 1 + 1 \\times 1 + 2 \\times 0}{1^2 + 1^2 + 0^2} = \\frac{4}{2} = 2,
\n$$<\/p>\n
$$
\n\\frac{\\langle \\mathbf{v}, \\mathbf{u}_2 \\rangle}{\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle} = \\frac{3 \\times 0 + 1 \\times 0 + 2 \\times 1}{0^2 + 0^2 + 1^2} = \\frac{2}{1} = 2.
\n$$<\/p>\n
\u3057\u305f\u304c\u3063\u3066:<\/p>\n
$$
\n\\text{proj}_W(\\mathbf{v}) = 2 \\times (1, 1, 0) + 2 \\times (0, 0, 1) = (2, 2, 2).
\n$$<\/p>\n
\u76f4\u4ea4\u6210\u5206 $\\mathbf{w}^\\perp = \\mathbf{v} - \\text{proj}_W(\\mathbf{v}) = (3, 1, 2) - (2, 2, 2) = (1, -1, 0)$\u3002<\/p>\n
**\u691c\u8a3c**: $\\langle \\mathbf{w}^\\perp, \\mathbf{u}_1 \\rangle = 1 \\times 1 + (-1) \\times 1 + 0 \\times 0 = 0$, $\\langle \\mathbf{w}^\\perp, \\mathbf{u}_2 \\rangle = 1 \\times 0 + (-1) \\times 0 + 0 \\times 1 = 0$\u3002$\\mathbf{w}^\\perp$ \u306f\u5b9f\u969b\u306b $W$ \u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u3002
\n```<\/p>\n
### 2.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)<\/p>\n
\u76f4\u4ea4\u5206\u89e3\u306e\u6700\u3082\u91cd\u8981\u306a\u5fdc\u7528\u306e\u4e00\u3064\u306f**\u4e3b\u6210\u5206\u5206\u6790(PCA, Principal Component Analysis)** \u3067\u3042\u308b$^{[24]}$\u3002PCA\u306f\u30c7\u30fc\u30bf\u306e\u5171\u5206\u6563\u884c\u5217\u3092\u5bfe\u89d2\u5316\u3057\u3001\u30c7\u30fc\u30bf\u306e\u5206\u6563\u304c\u6700\u3082\u5927\u304d\u3044\u65b9\u5411(\u4e3b\u6210\u5206)\u3092\u898b\u3064\u3051\u308b\u3002\u3053\u308c\u3089\u306e\u4e3b\u6210\u5206\u306f\u76f4\u4ea4\u57fa\u5e95\u3092\u306a\u3057\u3001\u5143\u306e\u30c7\u30fc\u30bf\u3092\u3053\u306e\u57fa\u5e95\u306b\u5c04\u5f71\u3059\u308b\u3053\u3068\u3067\u6b21\u5143\u524a\u6e1b\u3068\u30ce\u30a4\u30ba\u9664\u53bb\u3092\u9054\u6210\u3059\u308b\u3002<\/p>\n
\u5177\u4f53\u7684\u306b\u306f\u3001\u30c7\u30fc\u30bf\u884c\u5217 $X \\in \\mathbb{R}^{n \\times d}$(\u4e2d\u5fc3\u5316\u6e08\u307f)\u306e\u5171\u5206\u6563\u884c\u5217 $C = \\frac{1}{n-1} X^T X$ \u3092\u8003\u3048\u308b\u3002$C$ \u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb $\\{\\mathbf{e}_1, \\dots, \\mathbf{e}_d\\}$ \u306f\u76f4\u4ea4\u57fa\u5e95\u3092\u306a\u3057\u3001\u5bfe\u5fdc\u3059\u308b\u56fa\u6709\u5024 $\\lambda_1 \\geq \\lambda_2 \\geq \\dots \\geq \\lambda_d$ \u306f\u5404\u65b9\u5411\u306e\u5206\u6563\u3092\u8868\u3059\u3002\u30c7\u30fc\u30bf\u70b9 $\\mathbf{x}$ \u306e $k$ \u756a\u76ee\u306e\u4e3b\u6210\u5206\u4e0a\u3078\u306e\u5c04\u5f71\u306f $\\langle \\mathbf{x}, \\mathbf{e}_k \\rangle$ \u3067\u3042\u308b\u3002<\/p>\n
---<\/p>\n
## \u7b2c3\u7ae0 \u6700\u5c0f\u4e8c\u4e57\u6cd5 \u2014 \u5b58\u5728\u3057\u306a\u3044\u89e3\u3092\u898b\u3064\u3051\u308b\u65b9\u6cd5 (Chapter 3 Least Squares \u2014 How to Find a Solution That Doesn't Exist)<\/p>\n
### 3.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n
\u904e\u5270\u6c7a\u5b9a(Overdetermined)\u7dda\u5f62\u30b7\u30b9\u30c6\u30e0 $A\\mathbf{x} = \\mathbf{b}$(\u65b9\u7a0b\u5f0f\u306e\u6570 > \u672a\u77e5\u6570\u306e\u6570)\u306f\u901a\u5e38\u3001\u6b63\u78ba\u306a\u89e3\u3092\u6301\u305f\u306a\u3044\u3002**\u6700\u5c0f\u4e8c\u4e57\u6cd5(Least Squares Method)** \u306f\u6b8b\u5dee $\\|\\mathbf{b} - A\\mathbf{x}\\|$ \u306e $L_2$ \u30ce\u30eb\u30e0\u3092\u6700\u5c0f\u5316\u3059\u308b $\\hat{\\mathbf{x}}$ \u3092\u6c42\u3081\u308b$^{[5]}$\u3002<\/p>\n
```ad-theorem
\ntitle: \u5b9a\u7406 3.1 \u6b63\u898f\u65b9\u7a0b\u5f0f (Theorem 3.1 Normal Equations)
\n$A \\in \\mathbb{R}^{m \\times n}$($m > n$) \u304b\u3064 $\\mathbf{b} \\in \\mathbb{R}^m$ \u3068\u3059\u308b\u3002\u6700\u5c0f\u4e8c\u4e57\u554f\u984c $\\min_{\\mathbf{x}} \\|A\\mathbf{x} - \\mathbf{b}\\|^2$ \u306e\u89e3 $\\hat{\\mathbf{x}}$ \u306f\u6b21\u306e\u6b63\u898f\u65b9\u7a0b\u5f0f(Normal Equations)\u3092\u6e80\u305f\u3059:<\/p>\n
$$
\nA^T A \\hat{\\mathbf{x}} = A^T \\mathbf{b}.
\n\\tag{3.1}
\n$$<\/p>\n
$A$ \u306e\u5217\u304c\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308c\u3070 $A^T A$ \u306f\u53ef\u9006\u3067\u3042\u308a\u3001\u4e00\u610f\u306a\u89e3\u306f<\/p>\n
$$
\n\\hat{\\mathbf{x}} = (A^T A)^{-1} A^T \\mathbf{b}.
\n\\tag{3.2}
\n$$<\/p>\n
**\u8a3c\u660e**: \u30b3\u30b9\u30c8\u95a2\u6570 $J(\\mathbf{x}) = \\|A\\mathbf{x} - \\mathbf{b}\\|^2 = (A\\mathbf{x} - \\mathbf{b})^T (A\\mathbf{x} - \\mathbf{b})$ \u3092\u6700\u5c0f\u5316\u3059\u308b\u3002$J$ \u3092 $\\mathbf{x}$ \u306b\u3064\u3044\u3066\u5fae\u5206\u30570\u3068\u304a\u304f:<\/p>\n
$$
\n\\nabla J(\\mathbf{x}) = 2A^T (A\\mathbf{x} - \\mathbf{b}) = 0 \\implies A^T A \\mathbf{x} = A^T \\mathbf{b}.
\n$$<\/p>\n
$\\square$
\n```<\/p>\n
```ad-theorem
\ntitle: \u5b9a\u7406 3.2 \u5c04\u5f71\u3068\u3057\u3066\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5 (Theorem 3.2 Least Squares as Projection)
\n\u6700\u5c0f\u4e8c\u4e57\u89e3 $\\hat{\\mathbf{x}}$ \u306f $\\mathbf{b}$ \u3092 $A$ \u306e\u5217\u7a7a\u9593 $\\operatorname{Col}(A)$ \u306b\u76f4\u4ea4\u5c04\u5f71\u3057\u3066\u5f97\u3089\u308c\u308b:<\/p>\n
$$
\n\\hat{\\mathbf{y}} = A\\hat{\\mathbf{x}} = A (A^T A)^{-1} A^T \\mathbf{b} = P \\mathbf{b},
\n\\tag{3.3}
\n$$<\/p>\n
\u3053\u3053\u3067 $P = A (A^T A)^{-1} A^T$ \u306f $\\operatorname{Col}(A)$ \u4e0a\u3078\u306e\u76f4\u4ea4\u5c04\u5f71\u884c\u5217(Orthogonal Projection Matrix)\u3067\u3042\u308b\u3002$P$ \u306f\u51aa\u7b49(Idempotent, $P^2 = P$)\u304b\u3064\u5bfe\u79f0($P^T = P$)\u3067\u3042\u308b\u3002
\n```<\/p>\n
### 3.2 \u5e7e\u4f55\u5b66\u3068\u7a7a\u9593\u7684\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)<\/p>\n
\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306e\u5e7e\u4f55\u5b66\u7684\u610f\u5473\u306f\u975e\u5e38\u306b\u660e\u78ba\u3067\u3042\u308b: $\\mathbf{b}$ \u306f $A$ \u306e\u5217\u7a7a\u9593 $\\operatorname{Col}(A)$ \u306b\u5c5e\u3055\u306a\u3044\u3002\u6700\u9069\u306a\u8fd1\u4f3c\u306f $\\mathbf{b}$ \u3092 $\\operatorname{Col}(A)$ \u306b\u76f4\u4ea4\u5c04\u5f71\u3057\u3066\u5f97\u3089\u308c\u308b $\\hat{\\mathbf{y}}$ \u3067\u3042\u308b\u3002\u6b8b\u5dee $\\mathbf{r} = \\mathbf{b} - \\hat{\\mathbf{y}}$ \u306f\u5217\u7a7a\u9593\u306b\u5782\u76f4\u3067\u3042\u308a\u3001\u3057\u305f\u304c\u3063\u3066 $A^T \\mathbf{r} = \\mathbf{0}$ \u3092\u6e80\u305f\u3059\u3002<\/p>\n
### 3.3 \u30cf\u30fc\u30c9\u30b3\u30a2\u4f8b\u984c\u8a73\u7d30\u89e3\u8aac (Worked Example)<\/p>\n
```ad-example
\ntitle: \u4f8b\u984c 3.1 \u76f4\u7dda\u30d5\u30a3\u30c3\u30c6\u30a3\u30f3\u30b0 \u2014 \u6b63\u898f\u65b9\u7a0b\u5f0f\u306e\u624b\u8a08\u7b97 (Example 3.1 Line Fitting \u2014 Manual Solution of Normal Equations)
\n\u4e09\u3064\u306e\u30c7\u30fc\u30bf\u70b9 $(1, 1)$, $(2, 3)$, $(3, 2)$ \u304c\u4e0e\u3048\u3089\u308c\u305f\u3002\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u7528\u3044\u3066\u6700\u9069\u30d5\u30a3\u30c3\u30c6\u30a3\u30f3\u30b0\u76f4\u7dda $y = \\beta_0 + \\beta_1 x$ \u3092\u6c42\u3081\u3088\u3002<\/p>\n
**\u89e3**:<\/p>\n
**\u30b9\u30c6\u30c3\u30d7 1**: \u884c\u5217\u5f62\u5f0f\u3092\u69cb\u6210\u3002<\/p>\n
$$
\nA = \\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\\\ 1 & 3 \\end{bmatrix},\\quad
\n\\mathbf{b} = \\begin{bmatrix} 1 \\\\ 3 \\\\ 2 \\end{bmatrix},\\quad
\n\\mathbf{x} = \\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\end{bmatrix}.
\n$$<\/p>\n
**\u30b9\u30c6\u30c3\u30d7 2**: \u6b63\u898f\u65b9\u7a0b\u5f0f $A^T A \\hat{\\mathbf{x}} = A^T \\mathbf{b}$ \u3092\u69cb\u6210\u3002<\/p>\n
$$
\nA^T A = \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\end{bmatrix}
\n\\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\\\ 1 & 3 \\end{bmatrix}
\n= \\begin{bmatrix} 3 & 6 \\\\ 6 & 14 \\end{bmatrix},
\n$$<\/p>\n
$$
\nA^T \\mathbf{b} = \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\end{bmatrix}
\n\\begin{bmatrix} 1 \\\\ 3 \\\\ 2 \\end{bmatrix}
\n= \\begin{bmatrix} 6 \\\\ 13 \\end{bmatrix}.
\n$$<\/p>\n
\u6b63\u898f\u65b9\u7a0b\u5f0f:<\/p>\n
$$
\n\\begin{bmatrix} 3 & 6 \\\\ 6 & 14 \\end{bmatrix} \\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\end{bmatrix} = \\begin{bmatrix} 6 \\\\ 13 \\end{bmatrix}.
\n$$<\/p>\n
**\u30b9\u30c6\u30c3\u30d7 3**: \u9006\u884c\u5217\u3092\u7528\u3044\u3066\u89e3\u304f:<\/p>\n
$$
\n\\det = 3 \\times 14 - 6 \\times 6 = 42 - 36 = 6,
\n$$<\/p>\n
$$
\n(\\mathbf{A}^T \\mathbf{A})^{-1} = \\frac{1}{6} \\begin{bmatrix} 14 & -6 \\\\ -6 & 3 \\end{bmatrix},
\n$$<\/p>\n
$$
\n\\hat{\\mathbf{x}} = \\frac{1}{6} \\begin{bmatrix} 14 & -6 \\\\ -6 & 3 \\end{bmatrix} \\begin{bmatrix} 6 \\\\ 13 \\end{bmatrix} = \\frac{1}{6} \\begin{bmatrix} 84 - 78 \\\\ -36 + 39 \\end{bmatrix} = \\frac{1}{6} \\begin{bmatrix} 6 \\\\ 3 \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 0.5 \\end{bmatrix}.
\n$$<\/p>\n
\u3057\u305f\u304c\u3063\u3066\u6700\u9069\u76f4\u7dda\u306f $\\hat{y} = 1 + 0.5x$ \u3067\u3042\u308b\u3002<\/p>\n
**\u30b9\u30c6\u30c3\u30d7 4**: \u5c04\u5f71\u306e\u691c\u8a3c:<\/p>\n
$$
\n\\hat{\\mathbf{y}} = \\mathbf{A}\\hat{\\mathbf{x}} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\\\ 1 & 3 \\end{bmatrix} \\begin{bmatrix} 1 \\\\ 0.5 \\end{bmatrix} = \\begin{bmatrix} 1.5 \\\\ 2.0 \\\\ 2.5 \\end{bmatrix},
\n$$<\/p>\n
$$
\n\\mathbf{r} = \\mathbf{b} - \\hat{\\mathbf{y}} = \\begin{bmatrix} 1 \\\\ 3 \\\\ 2 \\end{bmatrix} - \\begin{bmatrix} 1.5 \\\\ 2.0 \\\\ 2.5 \\end{bmatrix} = \\begin{bmatrix} -0.5 \\\\ 1.0 \\\\ -0.5 \\end{bmatrix}.
\n$$<\/p>\n
$\\mathbf{A}^T \\mathbf{r} = \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\end{bmatrix} \\begin{bmatrix} -0.5 \\\\ 1.0 \\\\ -0.5 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}$ \u304c\u78ba\u8a8d\u3067\u304d\u3001\u6b8b\u5dee\u304c\u5217\u7a7a\u9593\u306b\u76f4\u4ea4\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002
\n```<\/p>\n
### 3.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)<\/p>\n
\u56f32\u306f\u6700\u5c0f\u4e8c\u4e57\u5c04\u5f71\u306e\u5e7e\u4f55\u5b66\u7684\u76f4\u89b3\u3092\u793a\u3059\u3002\u89b3\u6e2c\u30d9\u30af\u30c8\u30eb $\\mathbf{b}$ \u306f $\\mathbf{A}$ \u306e\u5217\u7a7a\u9593(\u30e2\u30c7\u30eb\u7a7a\u9593)\u306b\u5c5e\u3055\u306a\u3044\u3002\u6700\u5c0f\u4e8c\u4e57\u89e3 $\\hat{\\mathbf{x}}$ \u306f $\\mathbf{b}$ \u3092\u5217\u7a7a\u9593\u306b\u76f4\u4ea4\u5c04\u5f71\u3057\u3066 $\\hat{\\mathbf{y}}$ \u3092\u5f97\u3001\u3053\u308c\u306f\u5217\u7a7a\u9593\u306b\u304a\u3044\u3066 $\\mathbf{b}$ \u306b\u6700\u3082\u8fd1\u3044\u8fd1\u4f3c\u3067\u3042\u308b\u3002<\/p>\n
![](\"https:\/\/r2.wuhanqing.cn\/MyWebsiteFiles\/1-%E6%96%87%E7%AB%A0\/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0\/%E4%BB%8E%E7%82%B9%E7%A7%AF%E5%88%B0%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4%EF%BC%9A%E8%97%8F%E5%9C%A8%E5%BE%AE%E7%A7%AF%E5%88%86%E3%80%81%E4%BF%A1%E5%8F%B7%E4%B8%8EAI%E8%83%8C%E5%90%8E%E7%9A%84%E5%90%8C%E4%B8%80%E5%A5%97%E8%AF%AD%E8%A8%80\/Pictures\/02_least_squares_projection.png\")
<\/p>\n
**\u56f32: \u6700\u5c0f\u4e8c\u4e57\u6cd5\u306e\u76f4\u4ea4\u5c04\u5f71\u89e3\u91c8(Figure 2: Orthogonal Projection Interpretation of Least Squares).** $\\hat{\\mathbf{y}} = \\mathbf{A}\\hat{\\mathbf{x}}$ \u306f $\\mathbf{b}$ \u306e\u5217\u7a7a\u9593\u4e0a\u3078\u306e\u76f4\u4ea4\u5c04\u5f71\u3067\u3042\u308a\u3001\u6b8b\u5dee $\\mathbf{r} = \\mathbf{b} - \\hat{\\mathbf{y}}$ \u306f\u5217\u7a7a\u9593\u306b\u5782\u76f4\u3067\u3042\u308b\u3002<\/p>\n
\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306f\u30c7\u30fc\u30bf\u79d1\u5b66\u306e\u790e\u77f3\u3067\u3042\u308b$^{[25]}$: \u7dda\u5f62\u56de\u5e30(Linear Regression)\u306f\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u7528\u3044\u3066\u4e88\u6e2c\u5909\u6570\u3068\u5fdc\u7b54\u5909\u6570\u306e\u95a2\u4fc2\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b; \u30b7\u30b9\u30c6\u30e0\u540c\u5b9a(System Identification)\u306f\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u7528\u3044\u3066\u52d5\u7684\u30b7\u30b9\u30c6\u30e0\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u63a8\u5b9a\u3059\u308b; \u30ab\u30eb\u30de\u30f3\u30d5\u30a3\u30eb\u30bf(Kalman Filter)\u306e\u6e2c\u5b9a\u66f4\u65b0\u30b9\u30c6\u30c3\u30d7\u306f\u6700\u5c0f\u4e8c\u4e57\u554f\u984c\u3068\u3057\u3066\u89e3\u91c8\u3067\u304d\u308b\u3002\u3059\u3079\u3066\u306e\u5834\u5408\u306e\u6838\u5fc3\u306f\u540c\u3058\u3067\u3042\u308b: \u6b63\u78ba\u306a\u89e3\u304c\u5b58\u5728\u3057\u306a\u3044\u3068\u304d\u3001**\u76f4\u4ea4\u5c04\u5f71(Orthogonal Projection)** \u3092\u901a\u3058\u3066\u6700\u9069\u306a\u8fd1\u4f3c\u89e3\u3092\u898b\u3064\u3051\u308b\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n
---<\/p>\n
## \u7b2c4\u7ae0 \u95a2\u6570\u7a7a\u9593\u306e\u5185\u7a4d \u2014 \u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u3078\u306e\u62e1\u5f35 (Chapter 4 Inner Products in Function Spaces \u2014 Extension to Hilbert Space)<\/p>\n
### 4.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n
\u3053\u308c\u307e\u3067\u306e\u7ae0\u3067\u8b70\u8ad6\u3057\u305f\u5185\u7a4d\u306f\u3059\u3079\u3066\u6709\u9650\u6b21\u5143\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593 $\\mathbb{R}^n$ \u306b\u9650\u5b9a\u3055\u308c\u3066\u3044\u305f\u3002\u3057\u304b\u3057\u3001\u5185\u7a4d\u306e\u6982\u5ff5\u306f\u81ea\u7136\u306b\u7121\u9650\u6b21\u5143\u95a2\u6570\u7a7a\u9593\u3078\u3068\u4e00\u822c\u5316\u3067\u304d\u308b\u3002\u3053\u306e\u4e00\u822c\u5316\u306f\u95a2\u6570\u89e3\u6790\u5b66(Functional Analysis)\u306e\u4e2d\u6838\u3067\u3042\u308a\u3001\u7dda\u5f62\u4ee3\u6570\u3068\u4fe1\u53f7\u51e6\u7406\u3001\u91cf\u5b50\u529b\u5b66\u3092\u7d50\u3076\u67b6\u3051\u6a4b\u3067\u3042\u308b\u3002<\/p>\n
```ad-definition
\ntitle: \u5b9a\u7fa9 4.1 $L^2$ \u5185\u7a4d (Definition 4.1 $L^2$ Inner Product)
\n$f, g: [a, b] \\to \\mathbb{R}$ \u3092\u4e8c\u4e57\u53ef\u7a4d\u5206\u95a2\u6570(Square-Integrable Function)\u3001\u3059\u306a\u308f\u3061 $\\int_a^b [f(x)]^2 dx < \\infty$ \u3068\u3059\u308b\u3002\u305d\u306e\u5185\u7a4d\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b:\n\n$$\n\\langle f, g \\rangle = \\int_a^b f(x) g(x) \\, dx.\n\\tag{4.1}\n$$\n\n\u3053\u306e\u5185\u7a4d\u304b\u3089\u5c0e\u304b\u308c\u308b\u30ce\u30eb\u30e0\u306f\n\n$$\n\\|f\\| = \\sqrt{\\langle f, f \\rangle} = \\sqrt{\\int_a^b [f(x)]^2 \\, dx},\n\\tag{4.2}\n$$\n\n\u3067\u3042\u308a\u3001$L^2$ \u30ce\u30eb\u30e0($L^2$ Norm)\u3068\u547c\u3070\u308c\u3001\u7269\u7406\u7684\u306b\u306f\u4fe1\u53f7\u306e\u300c\u30a8\u30cd\u30eb\u30ae\u30fc(Energy)\u300d\u3068\u89e3\u91c8\u3055\u308c\u308b\u3053\u3068\u304c\u591a\u3044\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u7fa9 4.2 \u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593 (Definition 4.2 Hilbert Space)\n\u5b8c\u5099\u306a\u5185\u7a4d\u7a7a\u9593\u3092\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593(Hilbert Space)\u3068\u3044\u3046$^{[6][8]}$\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593 $\\mathcal{H}$ \u306f\u5185\u7a4d\u7a7a\u9593\u3067\u3042\u308a\u3001\u4efb\u610f\u306e\u30b3\u30fc\u30b7\u30fc\u5217(Cauchy Sequence)\u304c $\\mathcal{H}$ \u5185\u3067\u53ce\u675f\u3059\u308b\uff08\u3059\u306a\u308f\u3061\u7a7a\u9593\u304c\u5b8c\u5099\u3067\u3042\u308b\uff09\u3002\n\n\u6709\u9650\u6b21\u5143\u5185\u7a4d\u7a7a\u9593 $\\mathbb{R}^n$ \u306f\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306e\u7279\u5225\u306a\u5834\u5408\u3067\u3042\u308b\u3002\u7121\u9650\u6b21\u5143\u306e\u4f8b\u3068\u3057\u3066\u306f $L^2[a,b]$\uff08\u4e8c\u4e57\u53ef\u7a4d\u5206\u95a2\u6570\u7a7a\u9593\uff09\u3084 $\\ell^2$\uff08\u4e8c\u4e57\u53ef\u548c\u6570\u5217\u7a7a\u9593\uff09\u304c\u3042\u308b\u3002\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306e\u5b8c\u5099\u6027\u306b\u3088\u308a\u3001\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570(Fourier Series)\u306a\u3069\u306e\u7121\u9650\u7d1a\u6570\u5c55\u958b\u306e\u53ce\u675f\u304c\u4fdd\u8a3c\u3055\u308c\u308b\u3002\n```\n\n```ad-theorem\ntitle: \u5b9a\u7406 4.1 $L^2$ \u7a7a\u9593\u306b\u304a\u3051\u308b\u30b3\u30fc\u30b7\u30fc-\u30b7\u30e5\u30ef\u30eb\u30c4\u4e0d\u7b49\u5f0f (Theorem 4.1 Cauchy-Schwarz Inequality in $L^2$ Space)\n$L^2[a,b]$ \u5185\u306e\u4efb\u610f\u306e\u95a2\u6570 $f, g$ \u306b\u3064\u3044\u3066:\n\n$$\n\\left| \\int_a^b f(x) g(x) \\, dx \\right| \\leq \\sqrt{\\int_a^b [f(x)]^2 \\, dx} \\cdot \\sqrt{\\int_a^b [g(x)]^2 \\, dx}.\n\\tag{4.3}\n$$\n```\n\n### 4.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)\n\n\u95a2\u6570\u3092\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u898b\u308b\u9375\u306f\u3001\u300c\u70b9\u3054\u3068\u306e\u5bfe\u5fdc\u300d\u3068\u3044\u3046\u8003\u3048\u65b9\u3092\u7406\u89e3\u3059\u308b\u3053\u3068\u306b\u3042\u308b\u3002$\\mathbb{R}^n$ \u306b\u304a\u3044\u3066\u3001\u30d9\u30af\u30c8\u30eb $\\mathbf{v} = (v_1, \\dots, v_n)$ \u306e\u7b2c $i$ \u6210\u5206 $v_i$ \u306f\u7b2c $i$ \u5ea7\u6a19\u8ef8\u4e0a\u306e\u5024\u306b\u5bfe\u5fdc\u3059\u308b\u3002\u95a2\u6570\u7a7a\u9593\u3067\u306f\u3001\u5404 $x \\in [a,b]$ \u304c\u72ec\u7acb\u3057\u305f\u300c\u5ea7\u6a19\u8ef8\u300d\u306b\u5bfe\u5fdc\u3057\u3001\u95a2\u6570\u5024 $f(x)$ \u304c\u305d\u306e\u5ea7\u6a19\u8ef8\u4e0a\u306e\u6210\u5206\u3068\u306a\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u95a2\u6570 $f$ \u306f\u672c\u8cea\u7684\u306b\u53ef\u7b97\u7121\u9650\u500b\u306e\u6210\u5206\u3092\u6301\u3064\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\n\n\u4e8c\u3064\u306e\u95a2\u6570\u304c\u76f4\u4ea4\u3059\u308b\uff08$\\langle f, g \\rangle = 0$\uff09\u3053\u3068\u306f\u3001\u305d\u308c\u3089\u304c $L^2$ \u306e\u610f\u5473\u3067\u300c\u4e92\u3044\u306b\u76f8\u624b\u306e\u6210\u5206\u3092\u542b\u307e\u306a\u3044\u300d\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u3053\u306e\u6982\u5ff5\u306f\u4fe1\u53f7\u51e6\u7406\u306b\u304a\u3044\u3066\u6df1\u3044\u7269\u7406\u7684\u610f\u5473\u3092\u6301\u3064\uff1a\u76f4\u4ea4\u3059\u308b\u4fe1\u53f7\u306f\u540c\u4e00\u30c1\u30e3\u30cd\u30eb\u3067\u4e92\u3044\u306b\u5e72\u6e09\u3059\u308b\u3053\u3068\u306a\u304f\u4f1d\u9001\u3067\u304d\u308b\u3002\n\n### 4.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u984c 4.1 \u95a2\u6570\u7a7a\u9593\u306b\u304a\u3051\u308b\u76f4\u4ea4\u6027\u3068\u8ddd\u96e2\u306e\u6e2c\u5b9a (Example 4.1 Orthogonality and Distance in Function Space)\n\u533a\u9593 $[-1, 1]$ \u4e0a\u3067\u3001$f(x) = x$ \u3068 $g(x) = x^2$ \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u3059\u308b\u3002\u305d\u308c\u3089\u304c\u76f4\u4ea4\u3059\u308b\u304b\u3069\u3046\u304b\u3092\u5224\u5b9a\u3057\u3001\u305d\u308c\u305e\u308c\u306e\u30ce\u30eb\u30e0\u3068\u95a2\u6570\u9593\u8ddd\u96e2\u3092\u8a08\u7b97\u305b\u3088\u3002\n\n**\u89e3** (1) \u5185\u7a4d\u3092\u8a08\u7b97\u3059\u308b:\n\n$$\n\\langle f, g \\rangle = \\int_{-1}^{1} x \\cdot x^2 \\, dx = \\int_{-1}^{1} x^3 \\, dx = \\left[ \\frac{x^4}{4} \\right]_{-1}^{1} = \\frac{1}{4} - \\frac{1}{4} = 0.\n$$\n\n\u3057\u305f\u304c\u3063\u3066 $\\langle f, g \\rangle = 0$ \u3067\u3042\u308a\u3001$f$ \u3068 $g$ \u306f $[-1,1]$ \u4e0a\u3067\u76f4\u4ea4\u3059\u308b\u3002\u7406\u7531\u306f $x^3$ \u304c\u5947\u95a2\u6570\u3067\u3042\u308a\u3001\u5bfe\u79f0\u533a\u9593\u4e0a\u306e\u7a4d\u5206\u304c\u30bc\u30ed\u306b\u306a\u308b\u305f\u3081\u3067\u3042\u308b\u3002\n\n(2) \u30ce\u30eb\u30e0\u3092\u8a08\u7b97\u3059\u308b:\n\n$$\n\\|f\\| = \\sqrt{\\int_{-1}^{1} x^2 \\, dx} = \\sqrt{\\left[ \\frac{x^3}{3} \\right]_{-1}^{1}} = \\sqrt{\\frac{2}{3}} \\approx 0.8165,\n$$\n\n$$\n\\|g\\| = \\sqrt{\\int_{-1}^{1} x^4 \\, dx} = \\sqrt{\\left[ \\frac{x^5}{5} \\right]_{-1}^{1}} = \\sqrt{\\frac{2}{5}} \\approx 0.6325.\n$$\n\n(3) \u95a2\u6570\u9593\u8ddd\u96e2\u3092\u8a08\u7b97\u3059\u308b:\n\n$$\n\\|f - g\\|^2 = \\int_{-1}^{1} (x - x^2)^2 \\, dx = \\int_{-1}^{1} (x^2 - 2x^3 + x^4) \\, dx = \\frac{2}{3} + 0 + \\frac{2}{5} = \\frac{16}{15},\n$$\n\n\u3057\u305f\u304c\u3063\u3066 $d(f, g) = \\|f - g\\| = \\sqrt{16\/15} \\approx 1.0328$\u3002\n\n\u3053\u306e\u4f8b\u984c\u306f\u3001\u5947\u95a2\u6570\u3068\u5076\u95a2\u6570\u304c\u5bfe\u79f0\u533a\u9593\u4e0a\u3067\u81ea\u7136\u306b\u76f4\u4ea4\u3059\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u3053\u306e\u6027\u8cea\u306f\u30d5\u30fc\u30ea\u30a8\u89e3\u6790\u306b\u304a\u3044\u3066\u6975\u3081\u3066\u91cd\u8981\u3067\u3042\u308a\u3001\u6b63\u5f26\u57fa\u5e95\u3068\u4f59\u5f26\u57fa\u5e95\u306e\u9593\u306e\u76f4\u4ea4\u6027\u3092\u4fdd\u8a3c\u3059\u308b\u3002\n```\n\n### 4.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)\n\n\u95a2\u6570\u5185\u7a4d\u306e\u5de5\u5b66\u306b\u304a\u3051\u308b\u6700\u3082\u76f4\u63a5\u7684\u306a\u5fdc\u7528\u306f**\u30de\u30c3\u30c1\u30c9\u30d5\u30a3\u30eb\u30bf(Matched Filter)** \u3067\u3042\u308b\u3002\u30ec\u30fc\u30c0\u30fc\u304a\u3088\u3073\u901a\u4fe1\u30b7\u30b9\u30c6\u30e0\u306b\u304a\u3044\u3066\u3001\u53d7\u4fe1\u4fe1\u53f7 $r(t)$ \u3068\u9001\u4fe1\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8 $s(t)$ \u306e\u5185\u7a4d\n\n$$\n\\langle r, s \\rangle = \\int_{-\\infty}^{\\infty} r(t) s(t) \\, dt\n$$\n\n\u306f\u30bf\u30fc\u30b2\u30c3\u30c8\u306e\u6709\u7121\u3092\u691c\u51fa\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u308b\u3002\u30a8\u30b3\u30fc\u306b\u5bfe\u8c61\u7269\u304b\u3089\u306e\u53cd\u5c04\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u5834\u5408\u3001\u5185\u7a4d\u5024\u304c\u8457\u3057\u304f\u5897\u5927\u3059\u308b\u3002\u3053\u308c\u306f\u672c\u8cea\u7684\u306b\u95a2\u6570\u7a7a\u9593\u306b\u304a\u3051\u308b\u300c\u985e\u4f3c\u5ea6\u691c\u51fa(Similarity Detection)\u300d\u3067\u3042\u308b\u3002\n\n\u3055\u3089\u306b\u3001**\u30ab\u30fc\u30cd\u30eb\u6cd5(Kernel Methods)**$^{[22]}$ \u306e\u6838\u5fc3\u7684\u306a\u8003\u3048\u65b9\u306f\u3001\u30c7\u30fc\u30bf\u70b9\u3092\u518d\u751f\u6838\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593(RKHS: Reproducing Kernel Hilbert Space)\u306b\u5199\u50cf\u3057\u3001\u305d\u306e\u7121\u9650\u6b21\u5143\u7a7a\u9593\u3067\u5185\u7a4d\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u6697\u9ed9\u7684\u306b\u9ad8\u6b21\u5143\u7279\u5fb4\u5909\u63db\u3092\u5b9f\u73fe\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u3053\u308c\u306b\u3064\u3044\u3066\u306f\u7b2c12\u7ae0\u3067\u8a73\u3057\u304f\u8ad6\u3058\u308b\u3002\n\n---\n\n## \u7b2c5\u7ae0 \u4e09\u89d2\u95a2\u6570\u306e\u76f4\u4ea4\u6027 \u2014 \u5468\u6ce2\u6570\u9818\u57df\u306e\u57fa\u5e95\u95a2\u6570 (Chapter 5 Orthogonality of Trigonometric Functions \u2014 Basis Functions in the Frequency Domain)\n\n### 5.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)\n\n\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593 $L^2[-\\pi, \\pi]$ \u306b\u304a\u3044\u3066\u3001\u4e09\u89d2\u95a2\u6570\u7cfb\u306f\u91cd\u8981\u306a\u76f4\u4ea4\u57fa\u5e95\u3092\u69cb\u6210\u3059\u308b\u3002\u6b21\u306e\u95a2\u6570\u96c6\u5408\u3092\u8003\u3048\u308b:\n\n$$\n\\{1,\\ \\sin x,\\ \\cos x,\\ \\sin 2x,\\ \\cos 2x,\\ \\dots,\\ \\sin nx,\\ \\cos nx,\\ \\dots\\}.\n$$\n\n```ad-theorem\ntitle: \u5b9a\u7406 5.1 \u4e09\u89d2\u95a2\u6570\u306e\u76f4\u4ea4\u6027 (Theorem 5.1 Orthogonality of Trigonometric Functions)\n\u533a\u9593 $[-\\pi, \\pi]$ \u4e0a\u3067\u3001\u4e09\u89d2\u95a2\u6570\u7cfb\u306f\u4ee5\u4e0b\u306e\u76f4\u4ea4\u95a2\u4fc2\u3092\u6e80\u305f\u3059$^{[4]}$:\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin(mx) \\cos(nx) \\, dx = 0, \\quad \\forall m, n,\n\\tag{5.1}\n$$\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin(mx) \\sin(nx) \\, dx = 0, \\quad m \\neq n,\n\\tag{5.2}\n$$\n\n$$\n\\int_{-\\pi}^{\\pi} \\cos(mx) \\cos(nx) \\, dx = 0, \\quad m \\neq n.\n\\tag{5.3}\n$$\n\n\u540c\u4e00\u5468\u6ce2\u6570\u306e\u81ea\u5df1\u5185\u7a4d\u306f\u975e\u30bc\u30ed:\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin^2(nx) \\, dx = \\pi, \\quad\n\\int_{-\\pi}^{\\pi} \\cos^2(nx) \\, dx = \\pi.\n\\tag{5.4}\n$$\n\n**\u8a3c\u660e** \u3053\u308c\u3089\u306e\u95a2\u4fc2\u306f\u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u548c\u516c\u5f0f\u304b\u3089\u76f4\u63a5\u5c0e\u51fa\u3067\u304d\u308b\u3002\u4f8b\u3048\u3070 (5.2) \u306b\u3064\u3044\u3066:\n\n$$\n\\sin(mx)\\sin(nx) = \\frac{1}{2}[\\cos((m-n)x) - \\cos((m+n)x)].\n$$\n\n$m \\neq n$ \u306e\u3068\u304d\u3001$\\cos((m-n)x)$ \u3068 $\\cos((m+n)x)$ \u306e $[-\\pi, \\pi]$ \u4e0a\u306e\u7a4d\u5206\u306f\u3068\u3082\u306b\u30bc\u30ed\u3067\u3042\u308b\u3002$\\square$\n```\n\n### 5.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)\n\n\u4e09\u89d2\u95a2\u6570\u306e\u76f4\u4ea4\u6027\u306e\u5e7e\u4f55\u5b66\u7684\u610f\u5473\u306f\u3001\u7570\u306a\u308b\u5468\u6ce2\u6570\u306e\u6b63\u5f26\u6ce2\u3068\u4f59\u5f26\u6ce2\u304c $L^2$ \u7a7a\u9593\u306b\u304a\u3044\u3066\u4e92\u3044\u306b\u5782\u76f4\u3067\u3042\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u3053\u308c\u306f\u305d\u308c\u3089\u304c\u300c\u4fe1\u53f7\u300d\u3068\u3057\u3066\u4e92\u3044\u306b\u5e72\u6e09\u3057\u306a\u3044\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u2014\u2014\u3053\u308c\u3053\u305d\u304c\u5468\u6ce2\u6570\u5206\u5272\u591a\u91cd\u5316(Frequency-Division Multiplexing)\u306e\u6570\u5b66\u7684\u57fa\u76e4\u3067\u3042\u308b\u3002\n\n\u901a\u4fe1\u30b7\u30b9\u30c6\u30e0\u306b\u304a\u3044\u3066\u3001\u7570\u306a\u308b\u30e6\u30fc\u30b6\u306e\u30c7\u30fc\u30bf\u306f\u4e92\u3044\u306b\u76f4\u4ea4\u3059\u308b\u642c\u9001\u6ce2\u306b\u5909\u8abf\u3055\u308c\u3066\u540c\u6642\u306b\u4f1d\u9001\u3055\u308c\u3001\u53d7\u4fe1\u5074\u3067\u306f\u5185\u7a4d\u6f14\u7b97\u306b\u3088\u3063\u3066\u5404\u4fe1\u53f7\u3092\u5206\u96e2\u3067\u304d\u308b\u3002\u305f\u3068\u3048\u305d\u308c\u3089\u304c\u6642\u9593\u9818\u57df\u3067\u5b8c\u5168\u306b\u91cd\u306a\u3063\u3066\u3044\u3066\u3082\u554f\u984c\u306a\u3044\u3002\u3053\u306e\u539f\u7406\u306f\u73fe\u4ee3\u306e\u7121\u7dda\u901a\u4fe1\u306b\u304a\u3051\u308b**\u5468\u6ce2\u6570\u9818\u57df(Frequency Domain)**$^{[16]}$ \u89e3\u6790\u306e\u4e2d\u6838\u3092\u306a\u3059\u3002\n\n### 5.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u984c 5.1 \u4e09\u89d2\u95a2\u6570\u306e\u76f4\u4ea4\u6027\u306e\u624b\u8a08\u7b97\u306b\u3088\u308b\u691c\u8a3c (Example 5.1 Manual Verification of Trigonometric Orthogonality)\n$[-\\pi, \\pi]$ \u4e0a\u3067\u4ee5\u4e0b\u306e\u4e09\u7d44\u306e\u5185\u7a4d\u3092\u691c\u8a3c\u305b\u3088\u3002\n\n**\u30b1\u30fc\u30b9 A\uff1a$\\langle \\sin(2x), \\cos(3x) \\rangle$**\n\n$$\n\\langle \\sin(2x), \\cos(3x) \\rangle = \\int_{-\\pi}^{\\pi} \\sin(2x)\\cos(3x) \\, dx.\n$$\n\n\u7a4d\u548c\u516c\u5f0f $\\sin\\alpha\\cos\\beta = \\frac{1}{2}[\\sin(\\alpha+\\beta) + \\sin(\\alpha-\\beta)]$ \u3088\u308a:\n\n$$\n\\sin(2x)\\cos(3x) = \\frac{1}{2}[\\sin(5x) + \\sin(-x)] = \\frac{1}{2}[\\sin(5x) - \\sin(x)].\n$$\n\n$\\int_{-\\pi}^{\\pi} \\sin(kx) \\, dx = 0$ \u304c\u4efb\u610f\u306e\u6574\u6570 $k$ \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u306e\u3067:\n\n$$\n\\langle \\sin(2x), \\cos(3x) \\rangle = \\frac{1}{2} \\times 0 - \\frac{1}{2} \\times 0 = 0.\n$$\n\n**\u30b1\u30fc\u30b9 B\uff1a$\\langle \\sin(2x), \\sin(3x) \\rangle$**\n\n$\\sin\\alpha\\sin\\beta = \\frac{1}{2}[\\cos(\\alpha-\\beta) - \\cos(\\alpha+\\beta)]$ \u3088\u308a:\n\n$$\n\\sin(2x)\\sin(3x) = \\frac{1}{2}[\\cos(-x) - \\cos(5x)] = \\frac{1}{2}[\\cos(x) - \\cos(5x)].\n$$\n\n$\\int_{-\\pi}^{\\pi} \\cos(kx) \\, dx = 0$ \u304c $k \\neq 0$ \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u306e\u3067:\n\n$$\n\\langle \\sin(2x), \\sin(3x) \\rangle = \\frac{1}{2} \\times 0 - \\frac{1}{2} \\times 0 = 0.\n$$\n\n**\u30b1\u30fc\u30b9 C\uff1a$\\langle \\sin(2x), \\sin(2x) \\rangle$\uff08\u81ea\u5df1\u5185\u7a4d\uff09**\n\n\u500d\u89d2\u516c\u5f0f $\\sin^2\\theta = (1 - \\cos 2\\theta)\/2$ \u3092\u7528\u3044\u308b:\n\n$$\n\\langle \\sin(2x), \\sin(2x) \\rangle = \\int_{-\\pi}^{\\pi} \\frac{1 - \\cos(4x)}{2} \\, dx = \\frac{1}{2} \\cdot 2\\pi - 0 = \\pi.\n$$\n\n\u3053\u306e\u7d50\u679c\u306f $\\|\\sin(2x)\\| = \\sqrt{\\pi}$ \u3092\u610f\u5473\u3057\u3001\u3053\u308c\u304c\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570(Fourier Series)\u306e\u4fc2\u6570\u306e\u5206\u6bcd\u306b $\\pi$ \u304c\u73fe\u308c\u308b\u7406\u7531\u3067\u3042\u308b\u3002\n```\n\n### 5.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)\n\n**\u76f4\u4ea4\u5468\u6ce2\u6570\u5206\u5272\u591a\u91cd(OFDM: Orthogonal Frequency-Division Multiplexing)** \u306f\u73fe\u4ee3\u306e4G\/5G\u7121\u7dda\u901a\u4fe1\u306e\u4e2d\u6838\u6280\u8853\u3067\u3042\u308b$^{[16]}$\u3002\u9ad8\u901f\u30c7\u30fc\u30bf\u30b9\u30c8\u30ea\u30fc\u30e0\u3092\u8907\u6570\u306e\u4f4e\u901f\u30b5\u30d6\u30b9\u30c8\u30ea\u30fc\u30e0\u306b\u5206\u5272\u3057\u3001\u305d\u308c\u305e\u308c\u3092\u4e92\u3044\u306b\u76f4\u4ea4\u3059\u308b\u30b5\u30d6\u30ad\u30e3\u30ea\u30a2\u306b\u5909\u8abf\u3057\u3066\u4e26\u5217\u4f1d\u9001\u3059\u308b\u3002\u30b5\u30d6\u30ad\u30e3\u30ea\u30a2\u9593\u306e\u76f4\u4ea4\u6027\n\n$$\n\\int_0^T \\sin(2\\pi f_k t) \\cdot \\sin(2\\pi f_l t) \\, dt = 0, \\quad k \\neq l,\n$$\n\n\u306b\u3088\u308a\u3001\u53d7\u4fe1\u5074\u306f\u5185\u7a4d\u6f14\u7b97\u306b\u3088\u3063\u3066\u5404\u30b5\u30d6\u30ad\u30e3\u30ea\u30a2\u4fe1\u53f7\u3092\u5b8c\u5168\u306b\u5206\u96e2\u3067\u304d\u308b\u3002\u305f\u3068\u3048\u305d\u308c\u3089\u304c\u30b9\u30da\u30af\u30c8\u30eb\u4e0a\u3067\u5927\u304d\u304f\u91cd\u306a\u3063\u3066\u3044\u3066\u3082\u554f\u984c\u306a\u3044\u3002\u3053\u308c\u306b\u3088\u308a\u5468\u6ce2\u6570\u5229\u7528\u52b9\u7387\u304c\u5927\u5e45\u306b\u5411\u4e0a\u3059\u308b\u3002\n\n---\n\n## \u7b2c6\u7ae0 \u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3068\u30d5\u30fc\u30ea\u30a8\u5909\u63db \u2014 \u4e09\u89d2\u57fa\u5e95\u4e0a\u3078\u306e\u95a2\u6570\u306e\u5c04\u5f71 (Chapter 6 Fourier Series and Fourier Transform \u2014 Projection of Functions onto Trigonometric Bases)\n\n### 6.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)\n\n\u4e09\u89d2\u95a2\u6570\u7cfb\u306e\u76f4\u4ea4\u6027\u306b\u3088\u308a\u3001\u4efb\u610f\u306e\u5468\u671f\u95a2\u6570\u3092\u7570\u306a\u308b\u5468\u6ce2\u6570\u306e\u4e09\u89d2\u95a2\u6570\u306e\u7dda\u5f62\u7d50\u5408\u306b\u5206\u89e3\u3067\u304d\u308b\u3002\u3053\u306e\u5206\u89e3\u3092**\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570(Fourier Series)**$^{[11]}$ \u3068\u3044\u3046\u3002\n\n```ad-theorem\ntitle: \u5b9a\u7406 6.1 \u30d5\u30fc\u30ea\u30a8\u7d1a\u6570 (Theorem 6.1 Fourier Series)\n$f(t)$ \u3092 $2\\pi$ \u3092\u5468\u671f\u3068\u3059\u308b\u4e8c\u4e57\u53ef\u7a4d\u5206\u95a2\u6570\u3068\u3059\u308b\u3002\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u306f\n\n$$\nf(t) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} [a_n \\cos(nt) + b_n \\sin(nt)],\n\\tag{6.1}\n$$\n\n\u3067\u3042\u308a\u3001\u4fc2\u6570\u306f\u5185\u7a4d\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b:\n\n$$\na_0 = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\, dt,\n\\tag{6.2}\n$$\n\n$$\na_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\cos(nt) \\, dt = \\frac{\\langle f, \\cos(nt) \\rangle}{\\|\\cos(nt)\\|^2},\n\\tag{6.3}\n$$\n\n$$\nb_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\sin(nt) \\, dt = \\frac{\\langle f, \\sin(nt) \\rangle}{\\|\\sin(nt)\\|^2}.\n\\tag{6.4}\n$$\n\n\u5f0f (6.3)-(6.4) \u306f\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u672c\u8cea\u3092\u660e\u3089\u304b\u306b\u3059\u308b: \u305d\u308c\u3089\u306f\u95a2\u6570 $f$ \u306e\u5404\u4e09\u89d2\u57fa\u5e95\u3078\u306e\u5c04\u5f71\u4fc2\u6570\uff08\u5185\u7a4d\u3092\u57fa\u5e95\u306e\u30ce\u30eb\u30e0\u4e8c\u4e57\u3067\u5272\u3063\u305f\u3082\u306e\uff09\u3067\u3042\u308a\u3001\u6709\u9650\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306b\u304a\u3051\u308b\u76f4\u4ea4\u57fa\u5e95\u4e0a\u306e\u5ea7\u6a19\u8a08\u7b97\u3068\u5b8c\u5168\u306b\u4e00\u81f4\u3059\u308b\u3002\n\n\u5468\u671f $T \\to \\infty$ \u306e\u3068\u304d\u3001\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f**\u30d5\u30fc\u30ea\u30a8\u5909\u63db(Fourier Transform)**$^{[12]}$ \u3078\u3068\u79fb\u884c\u3059\u308b:\n\n$$\nX(f) = \\int_{-\\infty}^{\\infty} x(t) e^{-j2\\pi ft} \\, dt = \\langle x(t), e^{j2\\pi ft} \\rangle.\n\\tag{6.5}\n$$\n\n\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306f\u6642\u9593\u9818\u57df\u95a2\u6570 $x(t)$ \u3092\u8907\u7d20\u6307\u6570\u57fa\u5e95 $e^{j2\\pi ft}$ \u306b\u5c04\u5f71\u3057\u3001\u5468\u6ce2\u6570\u9818\u57df\u8868\u73fe $X(f)$ \u3092\u5f97\u308b\u3002\n```\n\n### 6.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)\n\n\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u5e7e\u4f55\u5b66\u7684\u672c\u8cea\u306f\u300c\u30d7\u30ed\u30fc\u30d6(Probe)\u300d\u306e\u8003\u3048\u65b9\u3067\u3042\u308b\uff1a\u7570\u306a\u308b\u5468\u6ce2\u6570\u306e\u8907\u7d20\u6307\u6570\u632f\u52d5\u3092\u30d7\u30ed\u30fc\u30d6\u3068\u3057\u3066\u7528\u3044\u3001\u89e3\u6790\u5bfe\u8c61\u306e\u4fe1\u53f7\u3068\u5185\u7a4d\u3092\u8a08\u7b97\u3059\u308b\u3002\u4fe1\u53f7\u304c\u305d\u306e\u5468\u6ce2\u6570\u6210\u5206\u3092\u542b\u3093\u3067\u3044\u308c\u3070\u5185\u7a4d\u5024\u306f\u5927\u304d\u304f\u306a\u308a\uff08\u30b9\u30da\u30af\u30c8\u30eb\u306b\u30d4\u30fc\u30af\u304c\u751f\u3058\u308b\uff09\u3001\u542b\u3093\u3067\u3044\u306a\u3051\u308c\u3070\u5185\u7a4d\u5024\u306f\u30bc\u30ed\u306b\u8fd1\u3065\u304f\u3002\u30b9\u30da\u30af\u30c8\u30eb\u56f3\u4e0a\u306e\u5404\u30d4\u30fc\u30af\u306f\u3001\u305d\u306e\u5468\u6ce2\u6570\u57fa\u5e95\u3078\u306e\u4fe1\u53f7\u306e\u5c04\u5f71\u5f37\u5ea6\u306b\u5bfe\u5fdc\u3059\u308b\u3002\n\n### 6.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u984c 6.1 \u5468\u671f\u65b9\u5f62\u6ce2\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b (Example 6.1 Fourier Series Expansion of a Periodic Square Wave)\n\u5468\u671f $2\\pi$ \u306e\u65b9\u5f62\u6ce2\n\n$$\nf(t) = \\begin{cases}\n1, & 0 < t < \\pi, \\\\\n-1, & -\\pi < t < 0,\n\\end{cases}\n$$\n\n\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u4fc2\u6570\u3092\u6c42\u3081\u3088\u3002\n\n**\u89e3** $f(t)$ \u306f\u5947\u95a2\u6570\u3067\u3042\u308b\u305f\u3081\u3001$a_0 = a_n = 0$\uff08\u4f59\u5f26\u4fc2\u6570\u306f\u3059\u3079\u3066\u30bc\u30ed\uff09\u3002$b_n$ \u306e\u307f\u3092\u8a08\u7b97\u3059\u308c\u3070\u3088\u3044\u3002\n\n$$\nb_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\sin(nt) \\, dt = \\frac{1}{\\pi} \\left( \\int_{-\\pi}^{0} (-\\sin(nt)) \\, dt + \\int_{0}^{\\pi} \\sin(nt) \\, dt \\right).\n$$\n\n\u7b2c\u4e00\u9805\u3092\u8a08\u7b97: $\\int_{-\\pi}^{0} -\\sin(nt) \\, dt = \\left[ \\frac{\\cos(nt)}{n} \\right]_{-\\pi}^{0} = \\frac{1}{n} - \\frac{\\cos(-n\\pi)}{n} = \\frac{1 - (-1)^n}{n}$\u3002\n\n\u7b2c\u4e8c\u9805\u3092\u8a08\u7b97: $\\int_{0}^{\\pi} \\sin(nt) \\, dt = \\left[ -\\frac{\\cos(nt)}{n} \\right]_{0}^{\\pi} = -\\frac{\\cos(n\\pi)}{n} + \\frac{1}{n} = \\frac{1 - (-1)^n}{n}$\u3002\n\n\u3057\u305f\u304c\u3063\u3066:\n\n$$\nb_n = \\frac{1}{\\pi} \\cdot \\frac{2[1 - (-1)^n]}{n} = \\begin{cases}\n\\dfrac{4}{n\\pi}, & n \\text{ \u304c\u5947\u6570\u306e\u3068\u304d}, \\\\[6pt]\n0, & n \\text{ \u304c\u5076\u6570\u306e\u3068\u304d}.\n\\end{cases}\n\\tag{6.6}\n$$\n\n\u6545\u306b\u65b9\u5f62\u6ce2\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u306f\n\n$$\nf(t) = \\frac{4}{\\pi} \\sum_{k=0}^{\\infty} \\frac{\\sin((2k+1)t)}{2k+1} = \\frac{4}{\\pi} \\left( \\sin t + \\frac{1}{3}\\sin 3t + \\frac{1}{5}\\sin 5t + \\cdots \\right).\n\\tag{6.7}\n$$\n\n\u6570\u5024\u691c\u8a3c: $t = \\pi\/2$ \u3068\u3059\u308b\u3068\u3001\u6700\u521d\u306e3\u9805\u306e\u8fd1\u4f3c\u306f\n\n$$\nf(\\pi\/2) \\approx \\frac{4}{\\pi} \\left( 1 - \\frac{1}{3} + \\frac{1}{5} \\right) = \\frac{52}{15\\pi} \\approx 1.103,\n$$\n\n\u3067\u3042\u308a\u3001\u771f\u5024 $1$ \u306b\u8fd1\u3044\u3002\u9805\u6570\u3092\u5897\u3084\u3059\u3068\u65b9\u5f62\u6ce2\u306b\u53ce\u675f\u3059\u308b\uff08\u30ae\u30d6\u30ba\u73fe\u8c61(Gibbs Phenomenon)\u306b\u3088\u308a\u4e0d\u9023\u7d9a\u70b9\u3067\u7d04 $9\\%$ \u306e\u30aa\u30fc\u30d0\u30fc\u30b7\u30e5\u30fc\u30c8\u304c\u751f\u3058\u308b\uff09\u3002\n```\n\n### 6.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)\n\n\u56f33\u306f\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u5178\u578b\u7684\u306a\u5fdc\u7528\u3092\u793a\u3057\u3066\u3044\u308b\u300250 Hz\u3001120 Hz\u3001260 Hz \u306e\u4e09\u3064\u306e\u5468\u6ce2\u6570\u6210\u5206\u3092\u542b\u3080\u30ce\u30a4\u30ba\u6df7\u3058\u308a\u306e\u4fe1\u53f7 $x(t)$ \u306f\u3001\u6642\u9593\u9818\u57df\u3067\u306f\u4e00\u898b\u4e71\u96d1\u306b\u898b\u3048\u308b\u3002\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u5f8c\u3001\u30b9\u30da\u30af\u30c8\u30eb\u56f3\u306f\u5bfe\u5fdc\u3059\u308b\u5468\u6ce2\u6570\u306b\u4e09\u3064\u306e\u660e\u78ba\u306a\u30d4\u30fc\u30af\u3092\u793a\u3059\u2014\u2014\u3053\u308c\u3089\u306f\u5404\u5468\u6ce2\u6570\u57fa\u5e95\u3078\u306e\u4fe1\u53f7\u306e\u5c04\u5f71\u5f37\u5ea6\u306b\u306f\u304b\u306a\u3089\u306a\u3044\u3002\n\n![](\"https:\/\/r2.wuhanqing.cn\/MyWebsiteFiles\/1-%E6%96%87%E7%AB%A0\/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0\/%E4%BB%8E%E7%82%B9%E7%A7%AF%E5%88%B0%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4%EF%BC%9A%E8%97%8F%E5%9C%A8%E5%BE%AE%E7%A7%AF%E5%88%86%E3%80%81%E4%BF%A1%E5%8F%B7%E4%B8%8EAI%E8%83%8C%E5%90%8E%E7%9A%84%E5%90%8C%E4%B8%80%E5%A5%97%E8%AF%AD%E8%A8%80\/Pictures\/03_fourier_decomposition.png\")
<\/p>\n
**\u56f33\uff1a\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u5468\u6ce2\u6570\u9818\u57df\u5c04\u5f71\u3002** \u4e0a\u56f3\u306f\u30ce\u30a4\u30ba\u6df7\u3058\u308a\u306e\u591a\u97f3\u4fe1\u53f7 $x(t) = 1.2\\sin(2\\pi\\cdot 50t) + 0.7\\sin(2\\pi\\cdot 120t) + 0.4\\sin(2\\pi\\cdot 260t) + \\eta(t)$ \u306e\u6642\u9593\u9818\u57df\u6ce2\u5f62\uff1b\u4e0b\u56f3\u306f\u632f\u5e45\u30b9\u30da\u30af\u30c8\u30eb\u3067\u3042\u308a\u300150\u3001120\u3001260 Hz \u306b\u9855\u8457\u306a\u30d4\u30fc\u30af\u304c\u73fe\u308c\u3066\u3044\u308b\u3002\u3053\u306e\u56f3\u306f main.py<\/a> \u5185\u306e `np.fft.rfft`\uff08\u96e2\u6563\u30d5\u30fc\u30ea\u30a8\u5909\u63db\uff09\u306b\u3088\u3063\u3066\u751f\u6210\u3055\u308c\u3001\u305d\u306e\u672c\u8cea\u306f\u6642\u9593\u9818\u57df\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u30d9\u30af\u30c8\u30eb\u3068\u8907\u7d20\u6307\u6570\u57fa\u5e95\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u8a08\u7b97\u3067\u3042\u308b\u3002<\/p>\n \u30d5\u30fc\u30ea\u30a8\u89e3\u6790\u306e\u5fdc\u7528\u306f\u5de5\u5b66\u306e\u3042\u3089\u3086\u308b\u5206\u91ce\u306b\u53ca\u3076\uff1aMP3 \u97f3\u58f0\u5727\u7e2e\u306f\u4eba\u9593\u306e\u8033\u306b\u654f\u611f\u3067\u306a\u3044\u9ad8\u5468\u6ce2\u6210\u5206\u3092\u5207\u308a\u6368\u3066\u308b\u3053\u3068\u3067\u30c7\u30fc\u30bf\u91cf\u3092\u524a\u6e1b\u3059\u308b\uff1bJPEG \u753b\u50cf\u5727\u7e2e\u306f\u96e2\u6563\u30b3\u30b5\u30a4\u30f3\u5909\u63db(DCT: Discrete Cosine Transform)$^{[18]}$ \u3092\u7528\u3044\u3066\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u3092\u5468\u6ce2\u6570\u57fa\u5e95\u306b\u5c04\u5f71\u3059\u308b\uff1b\u5fc3\u96fb\u56f3(ECG)\u4fe1\u53f7\u306e\u5468\u6ce2\u6570\u9818\u57df\u8a3a\u65ad\u306f\u30b9\u30da\u30af\u30c8\u30eb\u7279\u5fb4\u3092\u5229\u7528\u3057\u3066\u75c5\u7406\u30d1\u30bf\u30fc\u30f3\u3092\u8b58\u5225\u3059\u308b\u3002<\/p>\n ---<\/p>\n ## \u7b2c7\u7ae0 \u5468\u6ce2\u6570\u9818\u57df\u304b\u3089\u8907\u7d20\u5468\u6ce2\u6570\u9818\u57df\u3078 \u2014 \u30e9\u30d7\u30e9\u30b9\u5909\u63db\u3068Z\u5909\u63db (Chapter 7 From Frequency Domain to Complex Frequency Domain \u2014 Laplace and Z-Transforms)<\/p>\n ### 7.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n \u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306f\u4fe1\u53f7\u304c\u7d76\u5bfe\u53ef\u7a4d\u5206\u6761\u4ef6 $\\int_{-\\infty}^{\\infty} |f(t)|\\,dt < \\infty$ \u3092\u6e80\u305f\u3059\u3053\u3068\u3092\u8981\u6c42\u3059\u308b\u3002$f(t) = e^{2t}$\uff08$t \\geq 0$\uff09\u306e\u3088\u3046\u306a\u6307\u6570\u767a\u6563\u4fe1\u53f7\u306e\u5834\u5408\u3001\u305d\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306f $t$ \u306e\u5897\u52a0\u306b\u4f34\u3063\u3066\u767a\u6563\u3057\u3001\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u5185\u7a4d $\\langle f(t), e^{-j\\omega t} \\rangle$ \u306f\u53ce\u675f\u3057\u306a\u3044\u3002\u3053\u306e\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u306b\u306f\u3001\u63a2\u67fb\u57fa\u5e95\u3092\u7d14\u865a\u6307\u6570 $e^{-j\\omega t}$ \u304b\u3089\u5b9f\u90e8\u306b\u6e1b\u8870\u56e0\u5b50\u3092\u6301\u3064\u8907\u7d20\u6307\u6570 $e^{-st}$ \u3078\u3068\u4e00\u822c\u5316\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\u3053\u3053\u3067 $s = \\sigma + j\\omega$ \u3067\u3042\u308b\u3002\n\n```ad-definition\ntitle: \u5b9a\u7fa9 7.1 \u30e9\u30d7\u30e9\u30b9\u5909\u63db (Definition 7.1 Laplace Transform)\n$f(t)$ \u3092 $[0, \\infty)$ \u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570\u3068\u3059\u308b\u3002\u305d\u306e**\u30e9\u30d7\u30e9\u30b9\u5909\u63db(Laplace Transform)** \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b$^{[14]}$:\n\n$$F(s) = \\mathcal{L}\\{f(t)\\} = \\int_0^{\\infty} f(t) e^{-st}\\,dt, \\quad s = \\sigma + j\\omega \\in \\mathbb{C} \\tag{7.1}$$\n\n$s$ \u306e\u5b9f\u90e8 $\\sigma$ \u304c\u5341\u5206\u5927\u304d\u3044\u3068\u304d\u3001\u6e1b\u8870\u56e0\u5b50 $e^{-\\sigma t}$ \u306f $f(t)$ \u306e\u767a\u6563\u50be\u5411\u3092\u6291\u5236\u3057\u3001\u7a4d\u5206\u3092\u53ce\u675f\u3055\u305b\u308b\u3002(7.1) \u3092\u53ce\u675f\u3055\u305b\u308b $s$ \u306e\u5024\u306e\u96c6\u5408\u3092**\u53ce\u675f\u9818\u57df(ROC: Region of Convergence)** \u3068\u3044\u3046\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u7fa9 7.2 Z\u5909\u63db (Definition 7.2 Z-Transform)\n$x[n]$ \u3092 $\\mathbb{Z}$ \u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u96e2\u6563\u7cfb\u5217\u3068\u3059\u308b\u3002\u305d\u306e**Z\u5909\u63db(Z-Transform)** \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b$^{[15]}$:\n\n$$X(z) = \\mathcal{Z}\\{x[n]\\} = \\sum_{n=-\\infty}^{\\infty} x[n] z^{-n}, \\quad z = re^{j\\omega} \\in \\mathbb{C} \\tag{7.2}$$\n\nZ\u5909\u63db\u306f\u30e9\u30d7\u30e9\u30b9\u5909\u63db\u306e\u96e2\u6563\u9818\u57df\u306b\u304a\u3051\u308b\u5bfe\u5fdc\u7269\u3068\u898b\u306a\u305b\u308b\uff1a$z = e^{sT}$\uff08$T$ \u306f\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u5468\u671f\uff09\u3068\u304a\u304f\u3068\u3001$z$ \u5e73\u9762\u4e0a\u306e\u5358\u4f4d\u5186 $|z| = 1$ \u306f $s$ \u5e73\u9762\u4e0a\u306e\u865a\u8ef8 $s = j\\omega$ \u306b\u5bfe\u5fdc\u3059\u308b\u3002\n\n\u5185\u7a4d\u306e\u89b3\u70b9\u304b\u3089\u898b\u308b\u3068\u3001\u30e9\u30d7\u30e9\u30b9\u5909\u63db\u3068Z\u5909\u63db\u306f\u3068\u3082\u306b\u4fe1\u53f7\u3068\u8907\u7d20\u6307\u6570\u57fa\u5e95\u95a2\u6570\u3068\u306e\u5185\u7a4d\u3068\u3057\u3066\u7406\u89e3\u3067\u304d\u308b:\n\n$$\\mathcal{L}\\{f(t)\\} = \\langle f(t), e^{st} \\rangle, \\quad \\mathcal{Z}\\{x[n]\\} = \\langle x[n], z^n \\rangle$$\n\n\u3053\u3053\u3067\u57fa\u5e95\u95a2\u6570 $e^{st}$ \u3068 $z^n$ \u306f\u632f\u5e45\u6e1b\u8870\uff08$\\sigma$ \u307e\u305f\u306f $r$ \u306b\u3088\u308b\uff09\u3068\u4f4d\u76f8\u56de\u8ee2\uff08$\\omega$ \u306b\u3088\u308b\uff09\u306e\u4e8c\u3064\u306e\u81ea\u7531\u5ea6\u3092\u542b\u307f\u3001\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u57fa\u5e95\u3088\u308a\u3082\u8868\u73fe\u529b\u304c\u9ad8\u3044\u3002\n```\n\n### 7.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)\n\n\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u57fa\u5e95 $e^{-j\\omega t}$ \u306f\u8907\u7d20\u5e73\u9762\u4e0a\u306e\u5358\u4f4d\u5186\u4e0a\u306e\u7b49\u901f\u56de\u8ee2\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\u3001\u305d\u306e\u7d76\u5bfe\u5024\u306f\u5e38\u306b1\u3067\u3042\u308b\u3002\u767a\u6563\u4fe1\u53f7 $e^{2t}$ \u306e\u5834\u5408\u3001\u88ab\u7a4d\u5206\u95a2\u6570 $|e^{2t} \\cdot e^{-j\\omega t}| = e^{2t}$ \u306f $t$ \u306e\u5897\u52a0\u306b\u4f34\u3063\u3066\u767a\u6563\u3057\u3001\u7a4d\u5206\u306f\u6c7a\u3057\u3066\u53ce\u675f\u3057\u306a\u3044\u3002\n\n\u30e9\u30d7\u30e9\u30b9\u5909\u63db\u306e\u57fa\u5e95 $e^{-(\\sigma + j\\omega)t} = e^{-\\sigma t} e^{-j\\omega t}$ \u306f\u300c\u6e1b\u8870\u8abf\u6574\u3064\u307e\u307f\u300d$\\sigma$ \u3092\u8ffd\u52a0\u3057\u3066\u3044\u308b\u3002$\\sigma > 2$ \u306e\u3068\u304d\u3001$e^{-\\sigma t}$ \u306e\u6e1b\u8870\u7387\u304c $e^{2t}$ \u306e\u767a\u6563\u7387\u3092\u4e0a\u56de\u308a\u3001\u5185\u7a4d\u7a4d\u5206\u306f\u53ce\u675f\u3059\u308b\u3002\u8907\u7d20 $s$ \u5e73\u9762\u4e0a:<\/p>\n - **\u53ce\u675f\u9818\u57df(ROC)**\uff1a\u5909\u63db\u3092\u53ce\u675f\u3055\u305b\u308b $s$ \u5024\u306e\u9818\u57df\uff1b \u6975\u306e\u4f4d\u7f6e\u306f\u30b7\u30b9\u30c6\u30e0\u306e\u5b89\u5b9a\u6027\u3092\u76f4\u63a5\u6c7a\u5b9a\u3059\u308b\uff1a\u3059\u3079\u3066\u306e\u6975\u304c\u5de6\u534a\u5e73\u9762\uff08$\\text{Re}(s) < 0$\uff09\u306b\u3042\u308b\u3068\u304d\u30b7\u30b9\u30c6\u30e0\u306f\u5b89\u5b9a\uff1b\u3044\u305a\u308c\u304b\u306e\u6975\u304c\u53f3\u534a\u5e73\u9762\u306b\u3042\u308b\u3068\u304d\u30b7\u30b9\u30c6\u30e0\u306f\u767a\u6563\u3059\u308b\u3002\n\nZ\u5909\u63db\u306e\u5e7e\u4f55\u5b66\u7684\u89e3\u91c8\u3082\u540c\u69d8\u3067\u3042\u308b\uff1a$z = re^{j\\omega}$ \u306b\u304a\u3044\u3066\u3001$r$ \u306f\u632f\u5e45\u306e\u62e1\u5927\u7e2e\u5c0f\u3092\u5236\u5fa1\u3057\u3001$\\omega$ \u306f\u4f4d\u76f8\u56de\u8ee2\u3092\u5236\u5fa1\u3059\u308b\u3002\u53ce\u675f\u9818\u57df\u306f $|z| > R$\uff08\u53f3\u8fba\u7cfb\u5217\uff09\u307e\u305f\u306f $|z| < R$\uff08\u5de6\u8fba\u7cfb\u5217\uff09\u306e\u5186\u74b0\/\u5916\u90e8\u9818\u57df\u3068\u306a\u308b\u3002\u6975\u304c\u5358\u4f4d\u5186\u5185\u306b\u3042\u308b\u3068\u304d\u96e2\u6563\u30b7\u30b9\u30c6\u30e0\u306f\u5b89\u5b9a\u3067\u3042\u308b\u3002\n\n### 7.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u984c 7.1 \u767a\u6563\u95a2\u6570\u306e\u30e9\u30d7\u30e9\u30b9\u5909\u63db \u2014 \u6975\u3068\u53ce\u675f\u9818\u57df\u306e\u89e3\u6790 (Example 7.1 Laplace Transform of a Divergent Function \u2014 Pole and ROC Analysis)\n\n\u6307\u6570\u767a\u6563\u95a2\u6570 $f(t) = e^{2t}$\uff08$t \\geq 0$\uff09\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\u305d\u306e\u30e9\u30d7\u30e9\u30b9\u5909\u63db\u3092\u8a08\u7b97\u3057\u3001\u53ce\u675f\u9818\u57df\u3068\u6975\u3092\u89e3\u6790\u305b\u3088\u3002\n\n**\u89e3**\uff1a\u30e9\u30d7\u30e9\u30b9\u5909\u63db\u306e\u5b9a\u7fa9\u5f0f (7.1) \u306b\u4ee3\u5165\u3059\u308b:\n\n$$F(s) = \\int_0^{\\infty} e^{2t} \\cdot e^{-st}\\,dt = \\int_0^{\\infty} e^{-(s-2)t}\\,dt$$\n\n$a = s - 2 = (\\sigma - 2) + j\\omega$ \u3068\u304a\u304f:\n\n$$F(s) = \\int_0^{\\infty} e^{-at}\\,dt = \\left[-\\frac{1}{a}e^{-at}\\right]_{t=0}^{t=\\infty}$$\n\n$t \\to \\infty$ \u306e\u3068\u304d $e^{-at} \\to 0$ \u3068\u306a\u308b\u305f\u3081\u306b\u306f $\\text{Re}(a) > 0$\u3001\u3059\u306a\u308f\u3061 $\\text{Re}(s - 2) > 0$\u3001\u3064\u307e\u308a $\\sigma > 2$ \u304c\u5fc5\u8981\u3067\u3042\u308b\u3002\u3053\u306e\u6761\u4ef6\u4e0b\u3067:<\/p>\n $$F(s) = 0 - \\left(-\\frac{1}{a}\\right) = \\frac{1}{a} = \\frac{1}{s - 2}$$<\/p>\n \u3057\u305f\u304c\u3063\u3066:<\/p>\n $$\\mathcal{L}\\{e^{2t}\\} = \\frac{1}{s - 2}, \\quad \\text{ROC: } \\text{Re}(s) > 2, \\quad \\text{Pole: } s = 2$$<\/p>\n **\u5206\u6790**\uff1a\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306f $\\sigma = 0$ \u306b\u5bfe\u5fdc\u3059\u308b\u304c\u3001$s = j\\omega$ \u306e\u5b9f\u90e8\u306f0\u3067\u3042\u308a\u30012\u3088\u308a\u5c0f\u3055\u3044\u305f\u3081\u53ce\u675f\u9818\u57df\u5185\u306b\u306a\u3044\u2014\u2014\u3053\u308c\u304c $e^{2t}$ \u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u304c\u5b58\u5728\u3057\u306a\u3044\u7406\u7531\u3067\u3042\u308b\u3002\u30e9\u30d7\u30e9\u30b9\u5909\u63db\u306f\u5b9f\u90e8\u306e\u81ea\u7531\u5ea6 $\\sigma$ \u3092\u5c0e\u5165\u3059\u308b\u3053\u3068\u3067\u3001\u7a4d\u5206\u7d4c\u8def\u3092\u865a\u8ef8\u304b\u3089\u8907\u7d20\u5e73\u9762\u306e\u53f3\u534a\u5e73\u9762\u306b\u4e00\u822c\u5316\u3057\u3001\u767a\u6563\u4fe1\u53f7\u3092\u6271\u3048\u308b\u3088\u3046\u306b\u3059\u308b\u3002 ```ad-example \u96e2\u6563\u7cfb\u5217 $x[n] = (0.5)^n u[n]$ \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u3059\u308b\u3002\u3053\u3053\u3067 $u[n]$ \u306f\u5358\u4f4d\u30b9\u30c6\u30c3\u30d7\u95a2\u6570\uff08$n < 0$ \u30670\u3001$n \\geq 0$ \u30671\uff09\u3067\u3042\u308b\u3002\u305d\u306eZ\u5909\u63db\u3092\u8a08\u7b97\u3057\u3001\u53ce\u675f\u9818\u57df\u3068\u5b89\u5b9a\u6027\u3092\u89e3\u6790\u305b\u3088\u3002\n\n**\u89e3**\uff1aZ\u5909\u63db\u306e\u5b9a\u7fa9\u5f0f (7.2) \u306b\u4ee3\u5165\u3059\u308b:\n\n$$X(z) = \\sum_{n=0}^{\\infty} (0.5)^n z^{-n} = \\sum_{n=0}^{\\infty} (0.5 z^{-1})^n$$\n\n\u3053\u308c\u306f\u7b49\u6bd4\u7d1a\u6570\u3067\u3042\u308b\u3002$|0.5 z^{-1}| < 1$ \u3059\u306a\u308f\u3061 $|z| > 0.5$ \u306e\u3068\u304d\u7d1a\u6570\u306f\u53ce\u675f\u3059\u308b:<\/p>\n $$X(z) = \\frac{1}{1 - 0.5z^{-1}} = \\frac{z}{z - 0.5}, \\quad \\text{ROC: } |z| > 0.5$$<\/p>\n \u53ce\u675f\u9818\u57df\u306f\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u534a\u5f840.5\u306e\u5186\u306e\u5916\u90e8\u9818\u57df\u3067\u3042\u308b\u3002\u5358\u4f4d\u5186 $|z| = 1$ \u306f\u5b8c\u5168\u306b\u53ce\u675f\u9818\u57df\u5185\u306b\u3042\u308a\u3001\u3053\u306e\u7cfb\u5217\u306e\u96e2\u6563\u6642\u9593\u30d5\u30fc\u30ea\u30a8\u5909\u63db(DTFT: Discrete-Time Fourier Transform\u3001$z = e^{j\\omega}$ \u306b\u5bfe\u5fdc)\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u6975\u306f $z = 0.5$ \u306b\u3042\u308a\u3001\u5358\u4f4d\u5186\u5185\u90e8\u306b\u3042\u308b\u305f\u3081\u3001\u3053\u306e\u30b7\u30b9\u30c6\u30e0\u306f\u5b89\u5b9a\u3067\u3042\u308b\u3002 ### 7.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)<\/p>\n \u30e9\u30d7\u30e9\u30b9\u5909\u63db\u306f\u5236\u5fa1\u7406\u8ad6\u306e\u57fa\u76e4\u3067\u3042\u308b\u3002\u30d5\u30a3\u30fc\u30c9\u30d0\u30c3\u30af\u5236\u5fa1\u30b7\u30b9\u30c6\u30e0\u306b\u304a\u3044\u3066\u3001\u30b7\u30b9\u30c6\u30e0\u306e\u4f1d\u9054\u95a2\u6570 $H(s)$ \u306e\u6975\u306e\u4f4d\u7f6e\u304c\u5b89\u5b9a\u6027\u3092\u76f4\u63a5\u6c7a\u5b9a\u3059\u308b:<\/p>\n - \u3059\u3079\u3066\u306e\u6975\u304c\u5de6\u534a\u5e73\u9762\uff08$\\text{Re}(s) < 0$\uff09\u306b\u3042\u308b\uff1a\u30b7\u30b9\u30c6\u30e0\u306f\u5b89\u5b9a\u3001\u30a4\u30f3\u30d1\u30eb\u30b9\u5fdc\u7b54\u306f\u6307\u6570\u6e1b\u8870\uff1b\n- \u6975\u304c\u53f3\u534a\u5e73\u9762\uff08$\\text{Re}(s) > 0$\uff09\u306b\u5b58\u5728\u3059\u308b\uff1a\u30b7\u30b9\u30c6\u30e0\u306f\u767a\u6563\u3001\u30a4\u30f3\u30d1\u30eb\u30b9\u5fdc\u7b54\u306f\u6307\u6570\u5897\u52a0\uff1b Z\u5909\u63db\u306f\u30c7\u30b8\u30bf\u30eb\u4fe1\u53f7\u51e6\u7406\u306e\u4e2d\u6838\u3067\u3042\u308b\u3002\u30c7\u30b8\u30bf\u30eb\u30d5\u30a3\u30eb\u30bf\u306e\u5468\u6ce2\u6570\u5fdc\u7b54\u306f $H(z)$ \u306e\u5358\u4f4d\u5186\u4e0a\u306e\u5024\u306b\u3088\u3063\u3066\u6c7a\u5b9a\u3055\u308c\u3001\u5b89\u5b9a\u6027\u306f\u3059\u3079\u3066\u306e\u6975\u304c\u5358\u4f4d\u5186\u5185\u306b\u3042\u308b\u304b\u3069\u3046\u304b\u306b\u3088\u3063\u3066\u6c7a\u5b9a\u3055\u308c\u308b\u3002IIR\u30d5\u30a3\u30eb\u30bf\u8a2d\u8a08\u306f\u672c\u8cea\u7684\u306b ---<\/p>\n ## \u7b2c8\u7ae0 \u7573\u307f\u8fbc\u307f\u306e\u672c\u8cea \u2014 \u300c\u6ed1\u52d5\u3059\u308b\u5185\u7a4d\u300d (Chapter 8 The Essence of Convolution \u2014 \"Sliding Inner Product\")<\/p>\n ### 8.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n \u7573\u307f\u8fbc\u307f(Convolution)\u306f\u4fe1\u53f7\u51e6\u7406\u3001\u5236\u5fa1\u7406\u8ad6\u3001\u6df1\u5c64\u5b66\u7fd2\u306b\u304a\u3044\u3066\u6700\u3082\u4e2d\u6838\u7684\u306a\u6f14\u7b97\u306e\u4e00\u3064\u3067\u3042\u308b$^{[17]}$\u3002\u5185\u7a4d\u306e\u89b3\u70b9\u304b\u3089\u898b\u308b\u3068\u3001\u7573\u307f\u8fbc\u307f\u306e\u672c\u8cea\u306f**\u6ed1\u52d5\u7a93\u4e0a\u306e\u5185\u7a4d\u7cfb\u5217(Sliding Window Inner Product Sequence)** \u3067\u3042\u308b\u3002<\/p>\n ```ad-definition $$(f * g)(t) = \\int_{-\\infty}^{\\infty} f(\\tau) g(t - \\tau)\\,d\\tau \\tag{8.1}$$<\/p>\n \u96e2\u6563\u7cfb\u5217 $x, h: \\mathbb{Z} \\to \\mathbb{R}$ \u306b\u3064\u3044\u3066\u306f\u3001**\u96e2\u6563\u7573\u307f\u8fbc\u307f(Discrete Convolution)** \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b:<\/p>\n $$(x * h)[n] = \\sum_{k=-\\infty}^{\\infty} x[k]\\, h[n - k] \\tag{8.2}$$ ```ad-theorem $$(f * g)(t) = \\langle f(\\tau), g(t - \\tau) \\rangle = \\int f(\\tau) g(t - \\tau)\\,d\\tau \\tag{8.3}$$<\/p>\n \u3053\u3053\u3067\u53cd\u8ee2\u64cd\u4f5c $g(\\tau) \\to g(-\\tau)$ \u306f\u30b7\u30b9\u30c6\u30e0\u304c\u56e0\u679c\u6027\u3092\u6e80\u305f\u3059\u3053\u3068\u3092\u4fdd\u8a3c\u3059\u308b\u2014\u2014\u73fe\u5728\u306e\u51fa\u529b\u306f\u73fe\u5728\u304a\u3088\u3073\u904e\u53bb\u306e\u5165\u529b\u306e\u307f\u306b\u4f9d\u5b58\u3059\u308b\u3002 ```ad-definition $$(f \\star g)(t) = \\int_{-\\infty}^{\\infty} f(\\tau) g(\\tau + t)\\,d\\tau \\tag{8.4}$$<\/p>\n \u76f8\u4e92\u76f8\u95a2\u306f\u53cd\u8ee2\u64cd\u4f5c\u3092\u542b\u307e\u305a\u3001\u7570\u306a\u308b\u30aa\u30d5\u30bb\u30c3\u30c8\u306b\u304a\u3051\u308b\u4fe1\u53f7\u9593\u306e\u5185\u7a4d\u3092\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3002\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u30de\u30c3\u30c1\u30f3\u30b0\u3084\u985e\u4f3c\u5ea6\u691c\u51fa\u306b\u7528\u3044\u3089\u308c\u308b\u3002 ### 8.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)<\/p>\n \u7573\u307f\u8fbc\u307f\u306e\u5e7e\u4f55\u5b66\u7684\u30d7\u30ed\u30bb\u30b9\u306f\u56db\u3064\u306e\u30b9\u30c6\u30c3\u30d7\u306b\u5206\u89e3\u3067\u304d\u308b:<\/p>\n 1. **\u53cd\u8ee2(Flip)**\uff1a\u30ab\u30fc\u30cd\u30eb\u95a2\u6570 $g(\\tau)$ \u3092\u53cd\u8ee2\u3057\u3066 $g(-\\tau)$ \u3068\u3057\u3001\u6f14\u7b97\u304c\u56e0\u679c\u6027\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3059\u308b\uff1b $t$ \u306e\u5909\u5316\u306b\u4f34\u3044\u3001\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306f\u6642\u9593\u8ef8\u306b\u6cbf\u3063\u3066\u6ed1\u52d5\u3057\u3001\u5404\u4f4d\u7f6e\u3067\u4fe1\u53f7\u3068\u30ab\u30fc\u30cd\u30eb\u306e\u5185\u7a4d\u3092\u8a08\u7b97\u3059\u308b\u3002\u7573\u307f\u8fbc\u307f\u7d50\u679c $y(t)$ \u306f\u3001\u6ed1\u52d5\u4f4d\u7f6e\u306b\u5fdc\u3058\u305f\u5185\u7a4d\u5024\u306e\u5909\u5316\u66f2\u7dda\u3067\u3042\u308b\u3002\u5185\u7a4d\u5024\u304c\u5927\u304d\u3044\u4f4d\u7f6e\u306f\u3001\u4fe1\u53f7\u306e\u5c40\u6240\u90e8\u5206\u3068\u30ab\u30fc\u30cd\u30eb\u306e\u6ce2\u5f62\u304c\u6700\u3082\u985e\u4f3c\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3059\u2014\u2014\u3053\u308c\u304c\u307e\u3055\u306b**\u30de\u30c3\u30c1\u30c9\u30d5\u30a3\u30eb\u30bf(Matched Filter)** \u306e\u539f\u7406\u3067\u3042\u308b\u3002<\/p>\n \u753b\u50cf\u51e6\u7406\u306b\u304a\u3044\u3066\u3001\u4e8c\u6b21\u5143\u7573\u307f\u8fbc\u307f\u30ab\u30fc\u30cd\u30eb(Kernel)\u306f\u753b\u50cf\u4e0a\u3092\u6ed1\u52d5\u3057\u3001\u5404\u4f4d\u7f6e\u3067 $k \\times k$ \u8fd1\u508d\u3068\u30ab\u30fc\u30cd\u30eb\u306e\u4e8c\u6b21\u5143\u5185\u7a4d\u3092\u8a08\u7b97\u3057\u3001\u4e00\u679a\u306e\u300c\u5fdc\u7b54\u56f3(Feature Map)\u300d\u3092\u51fa\u529b\u3059\u308b\u3002\u5fdc\u7b54\u5024\u306e\u9ad8\u3044\u9818\u57df\u306f\u3001\u305d\u306e\u5c40\u6240\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u304c\u7573\u307f\u8fbc\u307f\u30ab\u30fc\u30cd\u30eb\u306e\u30d1\u30bf\u30fc\u30f3\u3068\u6700\u3082\u4e00\u81f4\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n ### 8.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)<\/p>\n ```ad-example \u5165\u529b\u7cfb\u5217 $x[n] = [1, 2, 3]$\uff08$n = 0, 1, 2$\uff09\u3068\u7573\u307f\u8fbc\u307f\u30ab\u30fc\u30cd\u30eb $h[n] = [0.5, 1, 0.5]$\uff08$n = 0, 1, 2$\uff09\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u3059\u308b\u3002\u7573\u307f\u8fbc\u307f $y[n] = (x * h)[n]$ \u3092\u8a08\u7b97\u305b\u3088\u3002<\/p>\n **\u89e3**\uff1a\u96e2\u6563\u7573\u307f\u8fbc\u307f\u516c\u5f0f (8.2) \u306b\u5f93\u3044\u3001\u4e00\u70b9\u305a\u3064\u8a08\u7b97\u3059\u308b:<\/p>\n $n = 0$: $n = 1$: $n = 2$: $n = 3$: $n = 4$: \u3057\u305f\u304c\u3063\u3066 $y[n] = [0.5, 2, 4, 4, 1.5]$\u3002$n = 2, 3$ \u3067\u7573\u307f\u8fbc\u307f\u5024\u304c\u6700\u5927\uff084\uff09\u3068\u306a\u308b\u3002\u3053\u306e\u3068\u304d\u5165\u529b\u7cfb\u5217 $[1, 2, 3]$ \u3068\u53cd\u8ee2\u30ab\u30fc\u30cd\u30eb $[0.5, 1, 0.5]$ \u306e\u91cd\u306a\u308a\u9818\u57df\u304c\u6700\u5927\u3068\u306a\u308a\u3001\u5185\u7a4d\u5024\u304c\u30d4\u30fc\u30af\u306b\u9054\u3059\u308b\u3002 ```ad-example Sobel \u6f14\u7b97\u5b50\u306f\u4e8c\u3064\u306e $3 \\times 3$ \u7573\u307f\u8fbc\u307f\u30ab\u30fc\u30cd\u30eb\u304b\u3089\u69cb\u6210\u3055\u308c\u3001\u305d\u308c\u305e\u308c\u6c34\u5e73\u65b9\u5411\u3068\u5782\u76f4\u65b9\u5411\u306e\u30a8\u30c3\u30b8\u3092\u691c\u51fa\u3059\u308b:<\/p>\n $$S_x = \\begin{bmatrix} 1 & 0 & -1 \\\\ 2 & 0 & -2 \\\\ 1 & 0 & -1 \\end{bmatrix}, \\quad S_y = \\begin{bmatrix} 1 & 2 & 1 \\\\ 0 & 0 & 0 \\\\ -1 & -2 & -1 \\end{bmatrix}$$<\/p>\n $3 \\times 3$ \u306e\u5c40\u6240\u753b\u50cf\u30d6\u30ed\u30c3\u30af\uff08\u8f1d\u5ea6\u5024\uff09\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u3059\u308b:<\/p>\n $$I = \\begin{bmatrix} 10 & 20 & 30 \\\\ 10 & 20 & 30 \\\\ 10 & 20 & 30 \\end{bmatrix}$$<\/p>\n \u3053\u306e\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u306f\u6c34\u5e73\u65b9\u5411\u306b\u660e\u308b\u3055\u306e\u52fe\u914d\uff08\u5de6\u304b\u3089\u53f3\u3078\u660e\u308b\u304f\u306a\u308b\uff09\u3092\u793a\u3057\u3001\u5782\u76f4\u65b9\u5411\u306b\u306f\u8f1d\u5ea6\u304c\u5747\u4e00\u3067\u3042\u308b\u3002<\/p>\n **\u89e3**\uff1aSobel X \u6f14\u7b97\u5b50\u3068\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u306e\u4e8c\u6b21\u5143\u5185\u7a4d\u3092\u8a08\u7b97\u3059\u308b:<\/p>\n $$G_x = \\sum_{i=1}^{3} \\sum_{j=1}^{3} S_x(i,j) \\cdot I(i,j)$$<\/p>\n $$= (1 \\times 10) + (0 \\times 20) + (-1 \\times 30) + (2 \\times 10) + (0 \\times 20) + (-2 \\times 30) + (1 \\times 10) + (0 \\times 20) + (-1 \\times 30)$$<\/p>\n $$= 10 + 0 - 30 + 20 + 0 - 60 + 10 + 0 - 30 = -80$$<\/p>\n Sobel Y \u6f14\u7b97\u5b50\u306e\u4e8c\u6b21\u5143\u5185\u7a4d\u3092\u8a08\u7b97\u3059\u308b:<\/p>\n $$G_y = (1 \\times 10) + (2 \\times 20) + (1 \\times 30) + (0 \\times 10) + (0 \\times 20) + (0 \\times 30) + (-1 \\times 10) + (-2 \\times 20) + (-1 \\times 30)$$<\/p>\n $$= 10 + 40 + 30 + 0 + 0 + 0 - 10 - 40 - 30 = 0$$<\/p>\n \u30a8\u30c3\u30b8\u5f37\u5ea6\u306f:<\/p>\n $$\\|\\nabla I\\| = \\sqrt{G_x^2 + G_y^2} = \\sqrt{(-80)^2 + 0^2} = 80$$<\/p>\n **\u5206\u6790**\uff1a$|G_x| = 80$ \u304c\u5927\u304d\u3044\u3053\u3068\u306f\u3001\u6c34\u5e73\u65b9\u5411\u306b\u9855\u8457\u306a\u8f1d\u5ea6\u5909\u5316\uff08\u5782\u76f4\u30a8\u30c3\u30b8\uff09\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u3059\uff1b$G_y = 0$ \u306f\u5782\u76f4\u65b9\u5411\u306e\u8f1d\u5ea6\u304c\u5747\u4e00\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3002Sobel \u30a8\u30c3\u30b8\u691c\u51fa\u306e\u672c\u8cea\u306f\u3001\u4e8c\u3064\u306e\u76f4\u4ea4\u3059\u308b\u7573\u307f\u8fbc\u307f\u30ab\u30fc\u30cd\u30eb\uff08\u5185\u7a4d\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\uff09\u3092\u753b\u50cf\u4e0a\u3067\u6ed1\u52d5\u3055\u305b\u3001\u5404\u30d4\u30af\u30bb\u30eb\u8fd1\u508d\u3068\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u306e\u4e8c\u6b21\u5143\u5185\u7a4d\u3092\u8a08\u7b97\u3057\u3001\u5185\u7a4d\u632f\u5e45\u304c\u5927\u304d\u3044\u4f4d\u7f6e\u3092\u30a8\u30c3\u30b8\u3068\u3057\u3066\u691c\u51fa\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002 ### 8.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)<\/p>\n > **\u56f34\uff1a\u6ed1\u52d5\u5185\u7a4d\u3068\u30de\u30c3\u30c1\u30c9\u30d5\u30a3\u30eb\u30bf(Matched Filter)\u3002** \u9752\u3044\u66f2\u7dda\u306f\u30ce\u30a4\u30ba\u6df7\u3058\u308a\u306e\u30e9\u30f3\u30c0\u30e0\u7cfb\u5217 $x[n]$\u3001\u8d64\u3044\u66f2\u7dda\u306f\u7573\u307f\u8fbc\u307f\u5fdc\u7b54\u3067\u3042\u308b\u3002\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u30d1\u30eb\u30b9 $h[n] = [0, 0.35, 1.0, 0.35, 0]$ \u304c\u6642\u9593\u8ef8\u306b\u6cbf\u3063\u3066\u6ed1\u52d5\u3057\u3001\u5404\u4f4d\u7f6e\u3067 $\\sum x[k]h[n-k]$ \u3092\u8a08\u7b97\u3059\u308b\u3002\u30aa\u30ec\u30f3\u30b8\u8272\u306e\u30de\u30fc\u30af\uff08$n \\approx 110, 265, 340$\uff09\u3067\u7573\u307f\u8fbc\u307f\u5024\u304c\u30d4\u30fc\u30af\u306b\u9054\u3057\u3066\u304a\u308a\u3001\u3053\u308c\u3089\u306e\u4f4d\u7f6e\u306e\u4fe1\u53f7\u5c40\u6240\u6ce2\u5f62\u304c\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u3068\u6700\u3082\u4e00\u81f4\u3059\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u73fe\u4ee3\u306e\u30ec\u30fc\u30c0\u30fc\u4fe1\u53f7\u6355\u6349\u306e\u6838\u5fc3\u539f\u7406\u306f\u3053\u306e\u6ed1\u52d5\u5c04\u5f71\u30e1\u30ab\u30cb\u30ba\u30e0\u306b\u7531\u6765\u3059\u308b\u3002<\/p>\n > **\u56f35\uff1a\u4e8c\u6b21\u5143\u7573\u307f\u8fbc\u307f\u306b\u3088\u308b\u30a8\u30c3\u30b8\u7279\u5fb4\u62bd\u51fa(Sobel Edge Detection)\u3002** Sobel \u6f14\u7b97\u5b50\u306f\u4e00\u5bfe\u306e\u76f4\u4ea4\u3059\u308b $3 \\times 3$ \u5fae\u5206\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u3067\u3042\u308a\u3001\u305d\u308c\u305e\u308c $x$ \u65b9\u5411\u3068 $y$ \u65b9\u5411\u306e\u8f1d\u5ea6\u52fe\u914d\u3092\u691c\u51fa\u3059\u308b\u3002\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u304c\u30b0\u30ec\u30fc\u30b9\u30b1\u30fc\u30eb\u753b\u50cf\u4e0a\u3092\u6ed1\u52d5\u3059\u308b\u3068\u304d\u3001\u5e73\u5766\u306a\u9818\u57df\u3067\u306f\u6b63\u8ca0\u306e\u5c04\u5f71\u304c\u4e92\u3044\u306b\u6253\u3061\u6d88\u3057\u5408\u3044\uff08\u5185\u7a4d\u304c\u30bc\u30ed\u306b\u8fd1\u3065\u304f\uff09\u3001\u30a8\u30c3\u30b8\u90e8\u5206\u3067\u306f\u30d4\u30af\u30bb\u30eb\u306e\u6bb5\u5dee\u306b\u3088\u3063\u3066\u5185\u7a4d\u632f\u5e45\u304c\u9855\u8457\u306b\u5897\u5927\u3059\u308b\u3002$\\|\\nabla I\\| = \\sqrt{G_x^2 + G_y^2}$ \u306b\u3088\u3063\u3066\u4e8c\u3064\u306e\u76f4\u4ea4\u6210\u5206\u3092\u7d71\u5408\u3059\u308b\u3053\u3068\u3067\u3001\u7269\u7406\u4e16\u754c\u306e\u30a8\u30c3\u30b8\u60c5\u5831\u3092\u62bd\u51fa\u3067\u304d\u308b\u3002\u3053\u308c\u306f\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30d3\u30b8\u30e7\u30f3\u306b\u304a\u3051\u308b\u7279\u5fb4\u62bd\u51fa\u306e\u57fa\u790e\u3067\u3042\u308b\u3002<\/p>\n ---<\/p>\n ## \u7b2c9\u7ae0 \u96e2\u6563\u30b3\u30b5\u30a4\u30f3\u5909\u63db\u3068JPEG\u5727\u7e2e (Chapter 9 Discrete Cosine Transform and JPEG Compression)<\/p>\n ### 9.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n \u96e2\u6563\u30b3\u30b5\u30a4\u30f3\u5909\u63db(DCT: Discrete Cosine Transform)\u306fJPEG\u753b\u50cf\u5727\u7e2e\u6a19\u6e96\u306e\u4e2d\u6838\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3042\u308b$^{[18][19]}$\u3002\u5185\u7a4d\u306e\u89b3\u70b9\u304b\u3089\u898b\u308b\u3068\u3001DCT\u306f\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u3092\u4e00\u7d44\u306e\u96e2\u6563\u30b3\u30b5\u30a4\u30f3\u57fa\u5e95\u95a2\u6570\u306b\u76f4\u4ea4\u5c04\u5f71\u3057\u3001\u7a7a\u9593\u9818\u57df\u306e\u30d4\u30af\u30bb\u30eb\u5024\u3092\u5468\u6ce2\u6570\u9818\u57df\u4fc2\u6570\u306b\u5909\u63db\u3059\u308b\u3002<\/p>\n ```ad-definition $$F(u, v) = \\frac{2}{N} C(u) C(v) \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} f(x, y) \\cos\\left[\\frac{(2x+1)u\\pi}{2N}\\right] \\cos\\left[\\frac{(2y+1)v\\pi}{2N}\\right] \\tag{9.1}$$<\/p>\n \u3053\u3053\u3067 $u, v = 0, 1, \\dots, N-1$ \u306f\u5468\u6ce2\u6570\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u3067\u3042\u308a\u3001\u6b63\u898f\u5316\u4fc2\u6570\u306f:<\/p>\n $$C(k) = \\begin{cases} 1\/\\sqrt{2}, & k = 0 \\\\ 1, & k \\neq 0 \\end{cases}$$ ```ad-theorem $$B_{u,v}(x, y) = \\frac{2}{N} C(u) C(v) \\cos\\left[\\frac{(2x+1)u\\pi}{2N}\\right] \\cos\\left[\\frac{(2y+1)v\\pi}{2N}\\right]$$<\/p>\n \u3053\u306e\u3068\u304d $\\{B_{u,v}\\}$ \u306f $\\mathbb{R}^{N \\times N}$ \u4e0a\u306e\u5b8c\u5099\u306a\u76f4\u4ea4\u57fa\u5e95\u3092\u69cb\u6210\u3057\u3001\u6b21\u3092\u6e80\u305f\u3059:<\/p>\n $$\\langle B_{u,v}, B_{u',v'} \\rangle = \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} B_{u,v}(x, y) B_{u',v'}(x, y) = \\delta_{u,u'} \\delta_{v,v'}$$<\/p>\n DCT\u4fc2\u6570 $F(u, v)$ \u306f\u307e\u3055\u306b\u753b\u50cf\u30d6\u30ed\u30c3\u30af $f$ \u306e\u57fa\u5e95\u95a2\u6570 $B_{u,v}$ \u3078\u306e\u5c04\u5f71\u3067\u3042\u308b:<\/p>\n $$F(u, v) = \\langle f, B_{u,v} \\rangle = \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} f(x, y) B_{u,v}(x, y) \\tag{9.2}$$ ```ad-theorem ### 9.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)<\/p>\n $8 \\times 8$ \u306e\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u306f64\u6b21\u5143\u7a7a\u9593\u4e2d\u306e\u30d9\u30af\u30c8\u30eb\u3068\u898b\u306a\u305b\u308b\u3002DCT\u57fa\u5e95\u95a2\u6570\u306f\u3053\u306e64\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u5b8c\u5099\u306a\u76f4\u4ea4\u57fa\u5e95\u3092\u69cb\u6210\u3059\u308b:<\/p>\n - **$B_{0,0}$\uff08DC\u57fa\u5e95\uff09**\uff1a\u5b9a\u6570\u95a2\u6570\u3001\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u306e\u5e73\u5747\u8f1d\u5ea6\u306b\u5bfe\u5fdc\uff1b \u753b\u50cf\u30d6\u30ed\u30c3\u30af\u30d9\u30af\u30c8\u30eb\u3092\u3053\u308c\u308964\u500b\u306e\u57fa\u5e95\u65b9\u5411\u306b\u5c04\u5f71\u3059\u308b\u3068\u300164\u500b\u306eDCT\u4fc2\u6570\u304c\u5f97\u3089\u308c\u308b\u3002\u81ea\u7136\u753b\u50cf\u306e\u5834\u5408\u3001\u5c04\u5f71\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u4f4e\u5468\u6ce2\u4fc2\u6570\uff08\u5de6\u4e0a\u9685\uff09\u306b\u9ad8\u5ea6\u306b\u96c6\u4e2d\u3057\u3001\u9ad8\u5468\u6ce2\u4fc2\u6570\uff08\u53f3\u4e0b\u9685\uff09\u306f\u30bc\u30ed\u306b\u8fd1\u3044\u3002JPEG\u5727\u7e2e\u306f\u91cf\u5b50\u5316\u306b\u3088\u3063\u3066\u5fae\u5c0f\u306a\u9ad8\u5468\u6ce2\u4fc2\u6570\u3092\u30bc\u30ed\u306b\u8a2d\u5b9a\u3057\u3001\u5c11\u6570\u306e\u4f4e\u5468\u6ce2\u4fc2\u6570\u306e\u307f\u3092\u4fdd\u6301\u3059\u308b\u3053\u3068\u3067\u5143\u306e\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u3092\u8fd1\u4f3c\u518d\u69cb\u6210\u3059\u308b\u3002<\/p>\n ### 9.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)<\/p>\n ```ad-example DCT\u306e\u5c04\u5f71\u672c\u8cea\u3092\u793a\u3059\u305f\u3081\u3001$N = 2$ \u306e\u30df\u30cb\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u3092\u8003\u3048\u308b\u3002$2 \\times 2$ DCT\u57fa\u5e95\u884c\u5217\u306f:<\/p>\n $$T = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}$$<\/p>\n $T$ \u306f\u76f4\u4ea4\u884c\u5217\u3067\u3042\u308a\u3001$T^T T = I$ \u3092\u6e80\u305f\u3059\u3002\u30b0\u30ec\u30fc\u30b9\u30b1\u30fc\u30eb\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u3059\u308b:<\/p>\n $$I = \\begin{bmatrix} 100 & 80 \\\\ 60 & 40 \\end{bmatrix}$$<\/p>\n \u4e8c\u6b21\u5143DCT\u306f\u884c\u5217\u4e57\u7b97\u306b\u3088\u3063\u3066\u5b9f\u73fe\u3067\u304d\u308b: $F = T \\cdot I \\cdot T^T$\u3002<\/p>\n **\u89e3**\uff1a<\/p>\n **\u30b9\u30c6\u30c3\u30d7 1**\uff1a$T \\cdot I$ \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n $$T \\cdot I = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} \\begin{bmatrix} 100 & 80 \\\\ 60 & 40 \\end{bmatrix} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix}$$<\/p>\n **\u30b9\u30c6\u30c3\u30d7 2**\uff1a$(T \\cdot I) \\cdot T^T$ \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n $$F = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix} \\cdot \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} = \\frac{1}{2} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}$$<\/p>\n $$= \\frac{1}{2} \\begin{bmatrix} 280 & 40 \\\\ 80 & 0 \\end{bmatrix} = \\begin{bmatrix} 140 & 20 \\\\ 40 & 0 \\end{bmatrix}$$<\/p>\n **\u30b9\u30c6\u30c3\u30d7 3**\uff1aDCT\u4fc2\u6570\u3092\u89e3\u91c8\u3059\u308b\u3002<\/p>\n - $F(0,0) = 140$\uff1aDC\u4fc2\u6570\u3001\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u306e\u5e73\u5747\u8f1d\u5ea6\u306b\u5bfe\u5fdc\u3002$(100+80+60+40)\/4 = 70$\u3001$N = 2$ \u3092\u4e57\u3058\u3066140\u3002 **\u91cd\u8981\u306a\u89b3\u5bdf**\uff1a$F(1,1) = 0$\u3001\u3059\u306a\u308f\u3061\u5bfe\u89d2\u65b9\u5411\u306e\u9ad8\u5468\u6ce2\u57fa\u5e95\u3078\u306e\u5c04\u5f71\u304c\u30bc\u30ed\u3067\u3042\u308b\u2014\u2014\u3053\u306e\u6210\u5206\u306f\u5b8c\u5168\u306b\u5207\u308a\u6368\u3066\u3066\u3082\u60c5\u5831\u304c\u5931\u308f\u308c\u306a\u3044\u3002\u3053\u308c\u304cJPEG\u5727\u7e2e\u306e\u6838\u5fc3\u539f\u7406\u3067\u3042\u308b\uff1a\u81ea\u7136\u753b\u50cf\u306e\u5927\u90e8\u5206\u306e\u9ad8\u5468\u6ce2DCT\u4fc2\u6570\u306f\u30bc\u30ed\u306b\u8fd1\u304f\u3001\u91cf\u5b50\u5316\u5f8c\u306b\u30bc\u30ed\u306b\u306a\u308b\u305f\u3081\u3001\u5927\u5e45\u306a\u5727\u7e2e\u304c\u5b9f\u73fe\u3067\u304d\u308b\u3002 ### 9.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)<\/p>\n JPEG\u5727\u7e2e\u306e\u6d41\u308c\u306f\u4ee5\u4e0b\u306e\u901a\u308a:<\/p>\n 1. **\u30d6\u30ed\u30c3\u30af\u5206\u5272**\uff1a\u753b\u50cf\u3092 $8 \\times 8$ \u306e\u30d6\u30ed\u30c3\u30af\u306b\u5206\u5272\u3059\u308b\uff1b \u5fa9\u53f7\u5074\u3067\u306f\u3001\u9006DCT\u5909\u63db\u306b\u3088\u3063\u3066\u753b\u50cf\u30d6\u30ed\u30c3\u30af\u3092\u518d\u69cb\u6210\u3059\u308b\u3002\u4eba\u9593\u306e\u76ee\u306b\u654f\u611f\u3067\u306a\u3044\u9ad8\u5468\u6ce2\u6210\u5206\u3092\u5207\u308a\u6368\u3066\u308b\u3053\u3068\u3067\u3001JPEG\u306f\u8996\u899a\u54c1\u8cea\u3092\u7dad\u6301\u3057\u3064\u3064\u753b\u50cf\u3092\u5143\u306e\u30b5\u30a4\u30ba\u306e $1\/10$ \u4ee5\u4e0b\u306b\u5727\u7e2e\u3067\u304d\u308b\u3002<\/p>\n DCT\u306f\u3055\u3089\u306b\u3001\u30d3\u30c7\u30aa\u5727\u7e2e\uff08MPEG\u3001H.264\/AVC\u3001HEVC\uff09\u3001\u97f3\u58f0\u5727\u7e2e\uff08MP3\u306b\u304a\u3051\u308bMDCT\u5909\u7a2e\uff09\u3001\u304a\u3088\u3073\u4fe1\u53f7\u51e6\u7406\u306b\u304a\u3051\u308b\u7121\u76f8\u95a2\u5316\u3068\u7279\u5fb4\u62bd\u51fa\u306b\u5e83\u304f\u5fdc\u7528\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n ---<\/p>\n ## \u7b2c10\u7ae0 \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db \u2014 \u30de\u30eb\u30c1\u30ec\u30be\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u5185\u7a4d (Chapter 10 Wavelet Transform \u2014 Multi-Resolution Inner Product)<\/p>\n ### 10.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)<\/p>\n \u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306f\u4fe1\u53f7\u3092\u7121\u9650\u306b\u5e83\u304c\u308b\u6b63\u5f26\u6ce2\u57fa\u5e95\u306b\u5c04\u5f71\u3057\u3001\u5927\u57df\u7684\u306a\u5468\u6ce2\u6570\u60c5\u5831\u3092\u5f97\u308b\u304c\u3001\u6642\u9593\u5b9a\u4f4d\u80fd\u529b\u3092\u5931\u3046\u2014\u2014\u30b9\u30da\u30af\u30c8\u30eb\u304b\u3089\u7279\u5b9a\u306e\u5468\u6ce2\u6570\u6210\u5206\u304c\u3044\u3064\u73fe\u308c\u305f\u304b\u3092\u77e5\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u3002\u97f3\u697d\u3001\u5730\u9707\u6ce2\u3001\u5fc3\u96fb\u56f3\u4fe1\u53f7\u306a\u3069\u306e\u975e\u5b9a\u5e38\u4fe1\u53f7\u306b\u3068\u3063\u3066\u3001\u3053\u306e\u300c\u6642\u9593\u76f2\u70b9\u300d\u306f\u6839\u672c\u7684\u306a\u6b20\u9665\u3067\u3042\u308b\u3002<\/p>\n ```ad-definition $$\\text{STFT}\\{f(t)\\}(\\tau, \\omega) = \\int_{-\\infty}^{\\infty} f(t) w(t - \\tau) e^{-j\\omega t}\\,dt$$<\/p>\n \u3057\u304b\u3057STFT\u306e\u7a93\u9577\u3092\u56fa\u5b9a\u3059\u308b\u3068\u3001\u6642\u9593\u5206\u89e3\u80fd $\\Delta t$ \u3068\u5468\u6ce2\u6570\u5206\u89e3\u80fd $\\Delta f$ \u306f\u30cf\u30a4\u30bc\u30f3\u30d9\u30eb\u30af\u306e\u4e0d\u78ba\u5b9a\u6027\u539f\u7406(Heisenberg Uncertainty Principle)\u306e\u5236\u7d04\u3092\u53d7\u3051\u308b$^{[16]}$:<\/p>\n $$\\Delta t \\cdot \\Delta f \\geq \\frac{1}{4\\pi} \\tag{10.1}$$ ```ad-definition $$\\psi_{a,b}(t) = \\frac{1}{\\sqrt{|a|}} \\psi\\left(\\frac{t - b}{a}\\right), \\quad a \\neq 0, \\; b \\in \\mathbb{R} \\tag{10.2}$$<\/p>\n \u3053\u3053\u3067 $a$ \u306f\u30b9\u30b1\u30fc\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf\uff08\u4f38\u7e2e\u3092\u5236\u5fa1\u3001\u5468\u6ce2\u6570\u306b\u5bfe\u5fdc\uff09\u3001$b$ \u306f\u5e73\u884c\u79fb\u52d5\u30d1\u30e9\u30e1\u30fc\u30bf\uff08\u4f4d\u7f6e\u3092\u5236\u5fa1\u3001\u6642\u9593\u306b\u5bfe\u5fdc\uff09\u3067\u3042\u308b\u3002\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u57fa\u5e95\u95a2\u6570\u306f\u6642\u9593\u9818\u57df\u306b\u304a\u3044\u3066**\u30b3\u30f3\u30d1\u30af\u30c8\u30b5\u30dd\u30fc\u30c8(Compact Support)** \u306e\u6027\u8cea\u3092\u6301\u3064\u2014\u2014\u6709\u9650\u533a\u9593\u5185\u3067\u306e\u307f\u975e\u30bc\u30ed\u2014\u2014\u3057\u305f\u304c\u3063\u3066\u81ea\u7136\u306b\u6642\u9593\u5b9a\u4f4d\u80fd\u529b\u3092\u5099\u3048\u308b\u3002 ```ad-definition $$W_f(a, b) = \\langle f, \\psi_{a,b} \\rangle = \\int_{-\\infty}^{\\infty} f(t) \\cdot \\frac{1}{\\sqrt{|a|}} \\psi^*\\left(\\frac{t - b}{a}\\right) dt \\tag{10.3}$$ ```ad-theorem - **\u5c0f\u30b9\u30b1\u30fc\u30eb $a$**\uff08\u9ad8\u5468\u6ce2\uff09\uff1a\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u306f\u5727\u7e2e\u3055\u308c\u3001\u6642\u9593\u5206\u89e3\u80fd\u304c\u9ad8\u304f\u5468\u6ce2\u6570\u5206\u89e3\u80fd\u304c\u4f4e\u3044\u3002\u904e\u6e21\u4fe1\u53f7\u306e\u89e3\u6790\u306b\u9069\u3059\u308b\uff1b \u3053\u306e**\u30de\u30eb\u30c1\u30ec\u30be\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u89e3\u6790(MRA: Multi-Resolution Analysis)** \u7279\u6027\u306f\u3001\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u3092\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3084STFT\u304b\u3089\u533a\u5225\u3059\u308b\u4e2d\u6838\u7684\u306a\u5229\u70b9\u3067\u3042\u308b\u3002 ### 10.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)<\/p>\n \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u306e\u5e7e\u4f55\u5b66\u7684\u30d7\u30ed\u30bb\u30b9\u306f\u3001\u7570\u306a\u308b\u30b5\u30a4\u30ba\u306e\u300c\u30d7\u30ed\u30fc\u30d6\u300d\u3092\u6642\u9593\u8ef8\u306b\u6cbf\u3063\u3066\u6ed1\u52d5\u3055\u305b\u308b\u3053\u3068\u3068\u3057\u3066\u7406\u89e3\u3067\u304d\u308b:<\/p>\n - **\u5927\u304d\u306a\u30d7\u30ed\u30fc\u30d6\uff08\u5927\u30b9\u30b1\u30fc\u30eb $a$\uff09**\uff1a\u5e83\u3044\u6642\u9593\u7bc4\u56f2\u3092\u30ab\u30d0\u30fc\u3057\u3001\u4fe1\u53f7\u306e\u9577\u671f\u7684\u306a\u30c8\u30ec\u30f3\u30c9\uff08\u4f4e\u5468\u6ce2\uff09\u3092\u611f\u77e5\u3059\u308b\u304c\u3001\u5909\u5316\u6642\u523b\u3092\u6b63\u78ba\u306b\u7279\u5b9a\u3067\u304d\u306a\u3044\uff1b \u5404\u4f4d\u7f6e $b$ \u306b\u304a\u3044\u3066\u3001\u4fe1\u53f7 $f(t)$ \u3068\u30d7\u30ed\u30fc\u30d6 $\\psi_{a,b}(t)$ \u306e\u5185\u7a4d $W_f(a, b)$ \u3092\u8a08\u7b97\u3059\u308b\u3002\u7d50\u679c\u306f**\u30b9\u30ab\u30ed\u30b0\u30e9\u30e0(Scalogram)** \u3068\u547c\u3070\u308c\u308b\u56f3\u3092\u69cb\u6210\u3057\u3001\u6a2a\u8ef8\u304c\u6642\u9593 $b$\u3001\u7e26\u8ef8\u304c\u30b9\u30b1\u30fc\u30eb $a$\uff08\u307e\u305f\u306f\u7b49\u4fa1\u5468\u6ce2\u6570\uff09\u3001\u8272\u306e\u6fc3\u6de1\u304c\u5185\u7a4d\u5f37\u5ea6\u3092\u8868\u3059\u3002<\/p>\n \u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3068\u306e\u5bfe\u6bd4\uff1a\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306f\u7121\u9650\u306b\u9577\u3044\u6b63\u5f26\u6ce2\u3067\u4fe1\u53f7\u5168\u4f53\u3092\u300c\u30de\u30c3\u30c1\u30f3\u30b0\u300d\u3057\u3001\u5927\u57df\u7684\u306a\u30b9\u30da\u30af\u30c8\u30eb\u3092\u5f97\u308b\uff1b\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u306f\u6709\u9650\u9577\u306e\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u3067\u4fe1\u53f7\u3092\u300c\u30b9\u30ad\u30e3\u30f3\u300d\u3057\u3001\u5404\u4f4d\u7f6e\u3067\u306e\u5c40\u6240\u7684\u306a\u30de\u30c3\u30c1\u30f3\u30b0\u5ea6\u3092\u8a18\u9332\u3059\u308b\u3053\u3068\u3067\u3001\u6642\u9593\u60c5\u5831\u3068\u5468\u6ce2\u6570\u60c5\u5831\u3092\u540c\u6642\u306b\u4fdd\u6301\u3059\u308b\u3002<\/p>\n ### 10.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)<\/p>\n ```ad-example Haar \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u306f\u6700\u3082\u5358\u7d14\u306a\u76f4\u4ea4\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u3067\u3042\u308a\u3001\u305d\u306e\u30b9\u30b1\u30fc\u30eb\u95a2\u6570 $\\phi(t)$ \u3068\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u95a2\u6570 $\\psi(t)$ \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b:<\/p>\n $$\\phi(t) = \\begin{cases} 1, & 0 \\leq t < 1 \\\\ 0, & \\text{otherwise} \\end{cases}, \\quad \\psi(t) = \\begin{cases} 1, & 0 \\leq t < 0.5 \\\\ -1, & 0.5 \\leq t < 1 \\\\ 0, & \\text{otherwise} \\end{cases}$$\n\n\u9577\u30558\u306e\u96e2\u6563\u4fe1\u53f7\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u3059\u308b:\n\n$$x = [4, 6, 10, 12, 8, 6, 5, 5]$$\n\nHaar \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5206\u89e3\u3092\u624b\u8a08\u7b97\u3067\u5b9f\u884c\u305b\u3088\u3002\n\n**\u89e3**\uff1a\n\n**\u30b9\u30c6\u30c3\u30d7 1\uff1a\u4e00\u6bb5\u5206\u89e3\u2014\u2014\u8fd1\u4f3c\u4fc2\u6570\u306e\u8a08\u7b97\u3002** \u8fd1\u4f3c\u4fc2\u6570\u306f\u30b9\u30b1\u30fc\u30eb\u95a2\u6570\u3068\u306e\u5185\u7a4d\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u308b\u3002\u3059\u306a\u308f\u3061\u96a3\u63a5\u4e8c\u70b9\u306e\u5e73\u5747\u5024:\n\n$$a_1 = \\frac{4+6}{2} = 5, \\quad a_2 = \\frac{10+12}{2} = 11, \\quad a_3 = \\frac{8+6}{2} = 7, \\quad a_4 = \\frac{5+5}{2} = 5$$\n\n\u8fd1\u4f3c\u4fc2\u6570\u30d9\u30af\u30c8\u30eb: $A^{(1)} = [5, 11, 7, 5]$\n\n**\u30b9\u30c6\u30c3\u30d7 2\uff1a\u4e00\u6bb5\u5206\u89e3\u2014\u2014\u8a73\u7d30\u4fc2\u6570\u306e\u8a08\u7b97\u3002** \u8a73\u7d30\u4fc2\u6570\u306f\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u95a2\u6570\u3068\u306e\u5185\u7a4d\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u308b\u3002\u3059\u306a\u308f\u3061\u96a3\u63a5\u4e8c\u70b9\u306e\u5dee\u306e\u534a\u5206:\n\n$$d_1 = \\frac{4-6}{2} = -1, \\quad d_2 = \\frac{10-12}{2} = -1, \\quad d_3 = \\frac{8-6}{2} = 1, \\quad d_4 = \\frac{5-5}{2} = 0$$\n\n\u8a73\u7d30\u4fc2\u6570\u30d9\u30af\u30c8\u30eb: $D^{(1)} = [-1, -1, 1, 0]$\n\n**\u30b9\u30c6\u30c3\u30d7 3\uff1a\u518d\u69cb\u6210\u306e\u691c\u8a3c\u3002** $A^{(1)}$ \u3068 $D^{(1)}$ \u304b\u3089\u5143\u306e\u4fe1\u53f7\u3092\u5b8c\u5168\u306b\u5fa9\u5143\u3067\u304d\u308b:\n\n$$x_1 = a_1 + d_1 = 5 + (-1) = 4, \\quad x_2 = a_1 - d_1 = 5 - (-1) = 6$$\n$$x_3 = a_2 + d_2 = 11 + (-1) = 10, \\quad x_4 = a_2 - d_2 = 11 - (-1) = 12$$\n$$x_5 = a_3 + d_3 = 7 + 1 = 8, \\quad x_6 = a_3 - d_3 = 7 - 1 = 6$$\n$$x_7 = a_4 + d_4 = 5 + 0 = 5, \\quad x_8 = a_4 - d_4 = 5 - 0 = 5$$\n\n\u518d\u69cb\u6210\u306f\u5b8c\u5168\u306b\u6b63\u3057\u3044\u3002\n\n**\u30b9\u30c6\u30c3\u30d7 4\uff1a\u4e8c\u6bb5\u5206\u89e3\u3002** \u8fd1\u4f3c\u4fc2\u6570 $A^{(1)} = [5, 11, 7, 5]$ \u306b\u5bfe\u3057\u3066\u3055\u3089\u306b Haar \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u3092\u65bd\u3059:\n\n$$a_1^{(2)} = \\frac{5+11}{2} = 8, \\quad a_2^{(2)} = \\frac{7+5}{2} = 6$$\n$$d_1^{(2)} = \\frac{5-11}{2} = -3, \\quad d_2^{(2)} = \\frac{7-5}{2} = 1$$\n\n\u4e8c\u6bb5\u8fd1\u4f3c: $A^{(2)} = [8, 6]$\u3001\u4e8c\u6bb5\u8a73\u7d30: $D^{(2)} = [-3, 1]$\n\n**\u91cd\u8981\u306a\u89b3\u5bdf**\uff1a\u5143\u306e\u4fe1\u53f7\u306f8\u500b\u306e\u6570\u5024\u3067\u4fdd\u5b58\u3055\u308c\u308b\u3002\u4e00\u6bb5\u5206\u89e3\u5f8c\u3001$A^{(1)}$\uff084\u5024\uff09+ $D^{(1)}$\uff084\u5024\uff09= 8\u5024\u3067\u3042\u308a\u3001\u5727\u7e2e\u306f\u3055\u308c\u3066\u3044\u306a\u3044\u3002\u3057\u304b\u3057\u3001\u7d76\u5bfe\u5024\u306e\u5c0f\u3055\u3044\u8a73\u7d30\u4fc2\u6570\uff08$d_4 = 0$ \u306a\u3069\uff09\u3092\u30bc\u30ed\u306b\u8a2d\u5b9a\u3059\u308c\u3070\u30017\u500b\u306e\u6709\u52b9\u5024\u306e\u307f\u3067\u4fdd\u5b58\u3067\u304d\u308b\u2014\u2014\u3053\u308c\u304c\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5727\u7e2e\u306e\u539f\u7406\u3067\u3042\u308b\u3002JPEG2000 \u306f\u307e\u3055\u306b\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\uff08CDF 9\/7 \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\uff09\u306b\u57fa\u3065\u3044\u3066\u304a\u308a\u3001JPEG\uff08DCT\uff09\u3088\u308a\u3082\u512a\u308c\u305f\u5727\u7e2e\u6027\u80fd\u3092\u5b9f\u73fe\u3057\u3001\u30d6\u30ed\u30c3\u30af\u6b6a\u307f\u3082\u767a\u751f\u3057\u306a\u3044\u3002\n```\n\n### 10.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)\n\n\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u89e3\u6790\u306f\u4fe1\u53f7\u51e6\u7406\u5206\u91ce\u3067\u5e45\u5e83\u304f\u5fdc\u7528\u3055\u308c\u3066\u3044\u308b:\n\n- **JPEG2000 \u753b\u50cf\u5727\u7e2e**\uff1aCDF 9\/7 \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u3092\u7528\u3044\u305f\u591a\u6bb5\u5206\u89e3\u306b\u3088\u308a\u3001JPEG\u306eDCT\u65b9\u5f0f\u3088\u308a\u3082\u5727\u7e2e\u7387\u304c\u9ad8\u304f\u3001\u30d6\u30ed\u30c3\u30af\u6b6a\u307f\u3082\u306a\u3044\uff1b\n- **\u5fc3\u96fb\u56f3(ECG)\u89e3\u6790**\uff1a\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u306fQRS\u6ce2\u7fa4\n\u3092\u6b63\u78ba\u306b\u7279\u5b9a\u3057\u3001\u4e0d\u6574\u8108\u691c\u51fa\u306b\u7528\u3044\u3089\u308c\u308b\uff1b\n- **\u5730\u9707\u4fe1\u53f7\u51e6\u7406**\uff1a\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u6642\u30b9\u30da\u30af\u30c8\u30eb\u306f\u5730\u9707\u6ce2\u306e\u5230\u9054\u6642\u9593\u3068\u5468\u6ce2\u6570\u6210\u5206\u3092\u540c\u6642\u306b\u660e\u3089\u304b\u306b\u3059\u308b\uff1b\n- **\u6df1\u5c64\u5b66\u7fd2\u306b\u304a\u3051\u308b\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u30cd\u30c3\u30c8\u30ef\u30fc\u30af**\uff1a\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u3092\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u524d\u7f6e\u7279\u5fb4\u62bd\u51fa\u5c64\u3068\u3057\u3066\u7528\u3044\u3001\u975e\u5b9a\u5e38\u4fe1\u53f7\u3092\u51e6\u7406\u3059\u308b\u3002\n\n---\n\n## \u7b2c11\u7ae0 \u81ea\u5df1\u6ce8\u610f\u30e1\u30ab\u30cb\u30ba\u30e0 \u2014 AI\u306e\u5185\u7a4d\u30a8\u30f3\u30b8\u30f3 (Chapter 11 Self-Attention Mechanism \u2014 AI's Inner Product Engine)\n\n### 11.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)\n\n\u73fe\u4ee3\u306e\u4eba\u5de5\u77e5\u80fd\u3001\u7279\u306bGPT\u3001BERT\u306a\u3069\u306e\u5927\u898f\u6a21\u8a00\u8a9e\u30e2\u30c7\u30eb(LLM: Large Language Model)\u306e\u57fa\u76e4\u8a08\u7b97\u306f\u3001\u307b\u307c\u3059\u3079\u3066\u5185\u7a4d\uff08\u30c9\u30c3\u30c8\u7a4d\uff09\u306b\u3088\u3063\u3066\u69cb\u6210\u3055\u308c\u3066\u3044\u308b\u3002Transformer\u30a2\u30fc\u30ad\u30c6\u30af\u30c1\u30e3\u306e\u4e2d\u6838\u2014\u2014**\u81ea\u5df1\u6ce8\u610f\u30e1\u30ab\u30cb\u30ba\u30e0(Self-Attention Mechanism)**\u2014\u2014\u306f\u3001\u672c\u8cea\u7684\u306b\u5927\u898f\u6a21\u3067\u4e26\u5217\u7684\u306a\u3001\u5b66\u7fd2\u53ef\u80fd\u306a\u30d9\u30af\u30c8\u30eb\u5185\u7a4d\u6f14\u7b97\u306e\u96c6\u5408\u3067\u3042\u308b$^{[18]}$\u3002\n\n```ad-definition\ntitle: \u5b9a\u7fa9 11.1 \u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u30c9\u30c3\u30c8\u7a4d\u6ce8\u610f (Definition 11.1 Scaled Dot-Product Attention)\n\u5165\u529b\u7cfb\u5217\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\u5404\u4f4d\u7f6e\u306e\u30c8\u30fc\u30af\u30f3\u306f\u4e09\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306b\u7dda\u5f62\u5c04\u5f71\u3055\u308c\u308b\uff1a\u30af\u30a8\u30ea\u30d9\u30af\u30c8\u30eb $Q$\u3001\u30ad\u30fc\u30d9\u30af\u30c8\u30eb $K$\u3001\u30d0\u30ea\u30e5\u30fc\u30d9\u30af\u30c8\u30eb $V$\u3002\u81ea\u5df1\u6ce8\u610f\u306e\u51fa\u529b\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b:\n\n$$\\text{Attention}(Q, K, V) = \\text{softmax}\\left(\\frac{QK^T}{\\sqrt{d_k}}\\right) V \\tag{11.1}$$\n\n\u3053\u3053\u3067 $Q \\in \\mathbb{R}^{n \\times d_k}$\u3001$K \\in \\mathbb{R}^{n \\times d_k}$\u3001$V \\in \\mathbb{R}^{n \\times d_v}$\u3001$n$ \u306f\u7cfb\u5217\u9577\u3001$d_k$ \u306f\u30af\u30a8\u30ea\/\u30ad\u30fc\u306e\u6b21\u5143\u3067\u3042\u308b\u3002\n```\n\n```ad-theorem\ntitle: \u547d\u984c 11.1 \u6b63\u898f\u5316\u5185\u7a4d\u3068\u3057\u3066\u306e\u6ce8\u610f\u91cd\u307f (Proposition 11.1 Attention Weights as Normalized Inner Products)\n\u884c\u5217 $QK^T$ \u306e\u7b2c $(i, j)$ \u8981\u7d20\u306f\u3001\u7b2c $i$ \u30af\u30a8\u30ea\u30d9\u30af\u30c8\u30eb\u3068\u7b2c $j$ \u30ad\u30fc\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u305d\u306e\u3082\u306e\u3067\u3042\u308b:\n\n$$(QK^T)_{ij} = \\langle Q_i, K_j \\rangle = Q_i \\cdot K_j = \\sum_{k=1}^{d_k} Q_{i,k} \\cdot K_{j,k} \\tag{11.2}$$\n\n\u3053\u306e\u5185\u7a4d\u5024\u304c\u5927\u304d\u3044\u307b\u3069\u3001\u7b2c $i$ \u30c8\u30fc\u30af\u30f3\u3068\u7b2c $j$ \u30c8\u30fc\u30af\u30f3\u306e\u95a2\u9023\u6027\u304c\u9ad8\u3044\u3053\u3068\u3092\u793a\u3059\u3002\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u56e0\u5b50 $1\/\\sqrt{d_k}$ \u306f\u3001\u5185\u7a4d\u5024\u304c\u6b21\u5143\u306e\u5897\u52a0\u306b\u4f34\u3063\u3066\u904e\u5927\u306b\u306a\u308b\u306e\u3092\u9632\u304e\u3001softmax\u306e\u52fe\u914d\u6d88\u5931\u3092\u56de\u907f\u3059\u308b\u3002Softmax\u306b\u3088\u308b\u6b63\u898f\u5316\u5f8c\u3001\u5185\u7a4d\u5024\u306f\u78ba\u7387\u91cd\u307f\u306b\u5909\u63db\u3055\u308c\u3001\u30d0\u30ea\u30e5\u30fc\u30d9\u30af\u30c8\u30eb $V$ \u306e\u91cd\u307f\u4ed8\u304d\u548c\u306b\u7528\u3044\u3089\u308c\u308b\u3002\n\n**\u30de\u30eb\u30c1\u30d8\u30c3\u30c9\u6ce8\u610f(Multi-Head Attention)** \u306f\u4e0a\u8a18\u306e\u30d7\u30ed\u30bb\u30b9\u3092 $h$ \u56de\uff08$h$ \u306f\u6ce8\u610f\u30d8\u30c3\u30c9\u6570\uff09\u4e26\u5217\u5b9f\u884c\u3057\u3001\u5404\u30d8\u30c3\u30c9\u304c\u7570\u306a\u308b\u5c04\u5f71\u90e8\u5206\u7a7a\u9593\u3092\u5b66\u7fd2\u3059\u308b:\n\n$$\\text{MultiHead}(Q, K, V) = \\text{Concat}(\\text{head}_1, \\dots, \\text{head}_h) W^O \\tag{11.3}$$\n\n\u3053\u3053\u3067 $\\text{head}_i = \\text{Attention}(Q W_i^Q, K W_i^K, V W_i^V)$ \u3067\u3042\u308b\u3002\n```\n\n### 11.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)\n\n\u81ea\u5df1\u6ce8\u610f\u30e1\u30ab\u30cb\u30ba\u30e0\u306f\u9ad8\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3044\u3066\u7cbe\u5de7\u306a\u300c\u5c04\u5f71-\u691c\u7d22\u300d\u64cd\u4f5c\u3092\u5b9f\u884c\u3059\u308b:\n\n1. **\u30af\u30a8\u30ea\u30d9\u30af\u30c8\u30eb $Q_i$**\uff1a\u300c\u8ab0\u304c\u79c1\u3068\u95a2\u9023\u3057\u3066\u3044\u308b\u304b\uff1f\u300d\u3068\u3044\u3046\u554f\u3044\u5408\u308f\u305b\u610f\u56f3\u3092\u7b26\u53f7\u5316\uff1b\n2. **\u30ad\u30fc\u30d9\u30af\u30c8\u30eb $K_j$**\uff1a\u300c\u79c1\u306f\u8ab0\u304b\u3001\u3069\u3093\u306a\u7279\u5fb4\u3092\u6301\u3063\u3066\u3044\u308b\u304b\uff1f\u300d\u3068\u3044\u3046\u8b58\u5225\u60c5\u5831\u3092\u7b26\u53f7\u5316\uff1b\n3. **\u5185\u7a4d $\\langle Q_i, K_j \\rangle$**\uff1a\u30af\u30a8\u30ea\u3068\u30ad\u30fc\u306e\u9ad8\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u985e\u4f3c\u5ea6\uff08\u30d9\u30af\u30c8\u30eb\u9593\u306e\u30b3\u30b5\u30a4\u30f3\u89d2\u306e\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u7248\uff09\u3092\u6e2c\u5b9a\uff1b\n4. **Softmax\u6b63\u898f\u5316**\uff1a\u985e\u4f3c\u5ea6\u3092\u78ba\u7387\u5206\u5e03\u306b\u5909\u63db\u3057\u3001\u30e2\u30c7\u30eb\u3092\u6700\u3082\u95a2\u9023\u6027\u306e\u9ad8\u3044\u30c8\u30fc\u30af\u30f3\u306b\u96c6\u4e2d\u3055\u305b\u308b\uff1b\n5. **\u91cd\u307f\u4ed8\u304d\u548c**\uff1a\u6ce8\u610f\u91cd\u307f\u306b\u5f93\u3063\u3066\u30d0\u30ea\u30e5\u30fc\u30d9\u30af\u30c8\u30eb\u304b\u3089\u6587\u8108\u60c5\u5831\u3092\u62bd\u51fa\u3059\u308b\u3002\n\nTransformer\u30e2\u30c7\u30eb\u5168\u4f53\u306f\u3001\u5de8\u5927\u306a**\u5fae\u5206\u53ef\u80fd\u5185\u7a4d\u30a8\u30f3\u30b8\u30f3(Differentiable Inner Product Engine)** \u3068\u898b\u306a\u305b\u308b\uff1a\u5404\u5c64\u3067\u5185\u7a4d\u6f14\u7b97\u3092\u5b9f\u884c\u3057\u3001\u9006\u4f1d\u64ad\u306b\u3088\u3063\u3066 $Q$\u3001$K$\u3001$V$ \u306e\u5c04\u5f71\u884c\u5217\u3092\u7d99\u7d9a\u7684\u306b\u8abf\u6574\u3057\u3001\u5185\u7a4d\u7d50\u679c\u304c\u30c7\u30fc\u30bf\u4e2d\u306e\u9577\u8ddd\u96e2\u4f9d\u5b58\u95a2\u4fc2\u3092\u6b63\u78ba\u306b\u6355\u6349\u3067\u304d\u308b\u3088\u3046\u306b\u3059\u308b\u3002\n\n### 11.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u984c 11.1 2\u30c8\u30fc\u30af\u30f3\u306e\u81ea\u5df1\u6ce8\u610f\u306e\u624b\u8a08\u7b97 (Example 11.1 Manual Calculation of Self-Attention for 2 Tokens)\n\n\u6975\u3081\u3066\u5358\u7d14\u306a\u7cfb\u5217\u3092\u8003\u3048\u308b\u3002\u4e8c\u3064\u306e\u30c8\u30fc\u30af\u30f3\u300c\u79c1\u300d\u3068\u300c\u611b\u300d\u306e\u307f\u3092\u542b\u3080\u3002\u57cb\u3081\u8fbc\u307f\u3068\u7dda\u5f62\u5c04\u5f71\u5f8c\uff08$d_k = 3$ \u3068\u3059\u308b\uff09:\n\n$$Q = \\begin{bmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{bmatrix}, \\quad\nK = \\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix}, \\quad\nV = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$$\n\n\u7b2c\u4e00\u884c\u304c\u300c\u79c1\u300d\u3001\u7b2c\u4e8c\u884c\u304c\u300c\u611b\u300d\u306b\u5bfe\u5fdc\u3059\u308b\u3002\n\n**\u89e3**\uff1a\n\n**\u30b9\u30c6\u30c3\u30d7 1\uff1a$QK^T$\uff08\u3059\u3079\u3066\u306e\u5185\u7a4d\u5bfe\uff09\u3092\u8a08\u7b97\u3059\u308b\u3002**\n\n$$QK^T = \\begin{bmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{bmatrix}\n\\begin{bmatrix} 1 & 0 \\\\ 1 & 1 \\\\ 0 & 1 \\end{bmatrix}$$\n\n\u8981\u7d20\u3054\u3068\u306b\u8a08\u7b97:\n- $(QK^T)_{11} = \\langle Q_1, K_1 \\rangle = 1 \\times 1 + 0 \\times 1 + 1 \\times 0 = 1$\n- $(QK^T)_{12} = \\langle Q_1, K_2 \\rangle = 1 \\times 0 + 0 \\times 1 + 1 \\times 1 = 1$\n- $(QK^T)_{21} = \\langle Q_2, K_1 \\rangle = 0 \\times 1 + 1 \\times 1 + 1 \\times 0 = 1$\n- $(QK^T)_{22} = \\langle Q_2, K_2 \\rangle = 0 \\times 0 + 1 \\times 1 + 1 \\times 1 = 2$\n\n$$QK^T = \\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\end{bmatrix}$$\n\n**\u30b9\u30c6\u30c3\u30d7 2\uff1a\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\uff08$\\sqrt{d_k} = \\sqrt{3} \\approx 1.732$ \u3067\u9664\u7b97\uff09\u3002**\n\n$$\\frac{QK^T}{\\sqrt{3}} = \\begin{bmatrix} 0.577 & 0.577 \\\\ 0.577 & 1.155 \\end{bmatrix}$$\n\n**\u30b9\u30c6\u30c3\u30d7 3\uff1aSoftmax\u6b63\u898f\u5316\uff08\u884c\u3054\u3068\uff09\u3002**\n\n\u7b2c\u4e00\u884c $[0.577, 0.577]$:\n$$e^{0.577} \\approx 1.781, \\quad \\text{sum} = 3.562$$\n$$\\text{softmax}_{11} = \\frac{1.781}{3.562} = 0.5, \\quad \\text{softmax}_{12} = \\frac{1.781}{3.562} = 0.5$$\n\n\u7b2c\u4e8c\u884c $[0.577, 1.155]$:\n$$e^{0.577} \\approx 1.781, \\quad e^{1.155} \\approx 3.174, \\quad \\text{sum} = 4.955$$\n$$\\text{softmax}_{21} = \\frac{1.781}{4.955} = 0.359, \\quad \\text{softmax}_{22} = \\frac{3.174}{4.955} = 0.641$$\n\n\u6ce8\u610f\u91cd\u307f\u884c\u5217:\n\n$$\\text{Weights} = \\begin{bmatrix} 0.5 & 0.5 \\\\ 0.359 & 0.641 \\end{bmatrix}$$\n\n**\u30b9\u30c6\u30c3\u30d7 4\uff1a\u91cd\u307f\u4ed8\u304d\u548c\u306b\u3088\u308a\u51fa\u529b\u3092\u5f97\u308b\u3002**\n\n$$\\text{Output} = \\text{Weights} \\cdot V = \\begin{bmatrix} 0.5 & 0.5 \\\\ 0.359 & 0.641 \\end{bmatrix}\n\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}$$\n\n- \u300c\u79c1\u300d\u306e\u65b0\u3057\u3044\u8868\u73fe: $0.5 \\times [1, 0] + 0.5 \\times [0, 1] = [0.5, 0.5]$\n- \u300c\u611b\u300d\u306e\u65b0\u3057\u3044\u8868\u73fe: $0.359 \\times [1, 0] + 0.641 \\times [0, 1] = [0.359, 0.641]$\n\n**\u91cd\u8981\u306a\u89b3\u5bdf**\uff1a\n- \u300c\u79c1\u300d\u306e\u6ce8\u610f\u306f\u4e8c\u3064\u306e\u30c8\u30fc\u30af\u30f3\u306b\u5747\u7b49\u306b\u5206\u5e03\u3057\u3066\u3044\u308b\uff08\u54040.5\uff09\u3002\u4e21\u8005\u3068\u306e\u5185\u7a4d\u304c\u540c\u3058\u3060\u304b\u3089\u3067\u3042\u308b\uff1b\n- \u300c\u611b\u300d\u306f\u300c\u79c1\u300d\uff080.359\uff09\u3088\u308a\u3082\u81ea\u5206\u81ea\u8eab\uff080.641\uff09\u306b\u6ce8\u76ee\u3057\u3066\u3044\u308b\u3002\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\uff082\uff09\u304c\u300c\u79c1\u300d\u3068\u306e\u5185\u7a4d\uff081\uff09\u3088\u308a\u5927\u304d\u3044\u305f\u3081\u3067\u3042\u308b\uff1b\n- \u51fa\u529b\u30d9\u30af\u30c8\u30eb\u306f\u30d0\u30ea\u30e5\u30fc\u30d9\u30af\u30c8\u30eb\u306e\u91cd\u307f\u4ed8\u304d\u7d50\u5408\u3067\u3042\u308a\u3001\u91cd\u307f\u306f\u5b8c\u5168\u306b\u5185\u7a4d\u306b\u3088\u3063\u3066\u6c7a\u5b9a\u3055\u308c\u308b\u2014\u2014\u3053\u308c\u304c\u300c\u5185\u7a4d\u3092\u901a\u3058\u305f\u6587\u8108\u8a8d\u8b58\u8868\u73fe\u300d\u306e\u4e2d\u6838\u30e1\u30ab\u30cb\u30ba\u30e0\u3067\u3042\u308b\u3002\n```\n\n### 11.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)\n\n\u81ea\u5df1\u6ce8\u610f\u30e1\u30ab\u30cb\u30ba\u30e0\u306e\u8a08\u7b97\u91cf\u306f\u7cfb\u5217\u9577 $n$ \u306b\u5bfe\u3057\u3066 $O(n^2)$ \u3067\u5897\u52a0\u3059\u308b\u3002GPT-4\u306a\u3069\u306e\u5927\u898f\u6a21\u30e2\u30c7\u30eb\uff08\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u9577\u306f\u6700\u5927128K\uff09\u3067\u306f\u3001\u5358\u4e00\u306e\u9806\u4f1d\u64ad\u306b\u6570\u5341\u5146\u56de\u306e\u5185\u7a4d\u6f14\u7b97\u304c\u5fc5\u8981\u3068\u306a\u308b\u3002\u8a08\u7b97\u3092\u9ad8\u901f\u5316\u3059\u308b\u305f\u3081\u3001\u696d\u754c\u3067\u306f\u69d8\u3005\u306a\u6700\u9069\u5316\u6280\u8853\u304c\u958b\u767a\u3055\u308c\u3066\u3044\u308b:\n\n- **Flash Attention**\uff1a\u30d6\u30ed\u30c3\u30af\u8a08\u7b97\u3068\u30e1\u30e2\u30ea\u6700\u9069\u5316\u306b\u3088\u308a\u3001GPU\u30e1\u30e2\u30ea\u306e\u8aad\u307f\u66f8\u304d\u3092\u524a\u6e1b\u3057\u3001\u6ce8\u610f\u8a08\u7b97\u30922\u20134\u500d\u9ad8\u901f\u5316\uff1b\n- **\u30b9\u30d1\u30fc\u30b9\u6ce8\u610f(Sparse Attention)**\uff1a\u4e00\u90e8\u306e\u30c8\u30fc\u30af\u30f3\u5bfe\u9593\u306e\u5185\u7a4d\u306e\u307f\u3092\u8a08\u7b97\u3057\uff08\u4f8b\uff1a\u5c40\u6240\u7a93 + \u5927\u57df\u30c8\u30fc\u30af\u30f3\uff09\u3001\u8a08\u7b97\u91cf\u3092 $O(n \\log n)$ \u306b\u524a\u6e1b\uff1b\n- **\u30de\u30eb\u30c1\u30af\u30a8\u30ea\u6ce8\u610f(MQA: Multi-Query Attention)**\uff1a\u8907\u6570\u306e\u30af\u30a8\u30ea\u30d8\u30c3\u30c9\u304c\u540c\u4e00\u306e\u30ad\u30fc\u5024\u30da\u30a2\u3092\u5171\u6709\u3057\u3001KV\u30ad\u30e3\u30c3\u30b7\u30e5\u30b5\u30a4\u30ba\u3092\u524a\u6e1b\uff1b\n- **\u7dda\u5f62\u6ce8\u610f(Linear Attention)**\uff1a\u30ab\u30fc\u30cd\u30eb\u6cd5\u3067softmax\u6ce8\u610f\u3092\u8fd1\u4f3c\u3057\u3001\u8a08\u7b97\u91cf\u3092 $O(n)$ \u306b\u524a\u6e1b\u3002\n\n\u3053\u308c\u3089\u306e\u6700\u9069\u5316\u306f\u672c\u8cea\u7684\u306b\u3001\u300c\u5185\u7a4d\u8a08\u7b97\u56de\u6570\u306e\u524a\u6e1b\u300d\u3068\u300c\u30e2\u30c7\u30eb\u306e\u8868\u73fe\u529b\u7dad\u6301\u300d\u306e\u9593\u3067\u6700\u9069\u306a\u30d0\u30e9\u30f3\u30b9\u3092\u63a2\u308b\u3082\u306e\u3067\u3042\u308b\u3002\n\n---\n\n## \u7b2c12\u7ae0 \u30ab\u30fc\u30cd\u30eb\u6cd5 \u2014 \u6697\u9ed9\u7684\u9ad8\u6b21\u5143\u5185\u7a4d (Chapter 12 Kernel Method \u2014 Implicit High-Dimensional Inner Product)\n\n### 12.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)\n\n\u4f4e\u6b21\u5143\u7a7a\u9593\u3067\u306f\u3001\u30c7\u30fc\u30bf\u306f\u3057\u3070\u3057\u3070\u7dda\u5f62\u5206\u96e2\u4e0d\u53ef\u80fd\u3067\u3042\u308b\u2014\u2014\u4f8b\u3048\u3070\u4e8c\u6b21\u5143\u5e73\u9762\u4e0a\u306e\u540c\u5fc3\u5186\u30c7\u30fc\u30bf\u306f\u4e00\u672c\u306e\u76f4\u7dda\u3067\u306f\u5206\u96e2\u3067\u304d\u306a\u3044\u3002\u5f93\u6765\u306e\u624b\u6cd5\u3067\u306f\u624b\u52d5\u3067\u9ad8\u6b21\u5143\u7279\u5fb4\u3092\u69cb\u7bc9\u3059\u308b\uff08\u4f8b\uff1a$x_1^2 + x_2^2$\uff09\u304c\u3001\u7279\u5fb4\u30a8\u30f3\u30b8\u30cb\u30a2\u30ea\u30f3\u30b0\u306e\u30b3\u30b9\u30c8\u306f\u6975\u3081\u3066\u9ad8\u3044\u3002**\u30ab\u30fc\u30cd\u30eb\u6cd5(Kernel Method)** \u306e\u6838\u5fc3\u7684\u306a\u8003\u3048\u65b9\u306f\uff1a\u9ad8\u6b21\u5143\u7a7a\u9593\u4e2d\u306e\u5ea7\u6a19\u3092\u660e\u793a\u7684\u306b\u8a08\u7b97\u305b\u305a\u3001\u9ad8\u6b21\u5143\u7a7a\u9593\u4e2d\u306e\u5185\u7a4d\u3092\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u3042\u308b$^{[22]}$\u3002\u3053\u306e\u6280\u6cd5\u3092**\u30ab\u30fc\u30cd\u30eb\u30c8\u30ea\u30c3\u30af(Kernel Trick)** \u3068\u3044\u3046\u3002\n\n```ad-definition\ntitle: \u5b9a\u7fa9 12.1 \u30ab\u30fc\u30cd\u30eb\u95a2\u6570 (Definition 12.1 Kernel Function)\n$\\phi: \\mathcal{X} \\to \\mathcal{H}$ \u3092\u5165\u529b\u7a7a\u9593\u304b\u3089\u9ad8\u6b21\u5143\uff08\u5834\u5408\u306b\u3088\u3063\u3066\u306f\u7121\u9650\u6b21\u5143\uff09\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u3078\u306e\u975e\u7dda\u5f62\u5199\u50cf\u3068\u3059\u308b\u3002\u30ab\u30fc\u30cd\u30eb\u95a2\u6570 $k: \\mathcal{X} \\times \\mathcal{X} \\to \\mathbb{R}$ \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b:\n\n$$k(x, y) = \\langle \\phi(x), \\phi(y) \\rangle_{\\mathcal{H}} \\tag{12.1}$$\n\n\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306e\u7cbe\u5999\u306a\u70b9\u306f\uff1a$\\phi$ \u306e\u5177\u4f53\u7684\u306a\u5f62\u5f0f\u3092\u77e5\u308b\u5fc5\u8981\u304c\u306a\u304f\u3001$k(x, y)$ \u304c**Mercer\u6761\u4ef6**\uff08\u5bfe\u79f0\u304b\u3064\u534a\u6b63\u5b9a\u5024\uff09\u3092\u6e80\u305f\u305b\u3070\u3001\u305d\u308c\u304c\u4f55\u3089\u304b\u306e\u518d\u751f\u6838\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593(RKHS: Reproducing Kernel Hilbert Space)\u306b\u304a\u3051\u308b\u5185\u7a4d\u306b\u5bfe\u5fdc\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u7fa9 12.2 \u4e00\u822c\u7684\u306a\u30ab\u30fc\u30cd\u30eb\u95a2\u6570 (Definition 12.2 Common Kernel Functions)\n\u3088\u304f\u7528\u3044\u3089\u308c\u308b\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u306b\u306f\u4ee5\u4e0b\u304c\u3042\u308b:\n\n- **\u7dda\u5f62\u30ab\u30fc\u30cd\u30eb(Linear Kernel)**\uff1a$k(x, y) = x^T y$\uff08\u5143\u306e\u7a7a\u9593\u306b\u304a\u3051\u308b\u5185\u7a4d\uff09\uff1b\n- **\u591a\u9805\u5f0f\u30ab\u30fc\u30cd\u30eb(Polynomial Kernel)**\uff1a$k(x, y) = (x^T y + c)^d$\uff08$d$ \u6b21\u591a\u9805\u5f0f\u7279\u5fb4\u7a7a\u9593\u306b\u5bfe\u5fdc\uff09\uff1b\n- **\u30ac\u30a6\u30b9\u52d5\u5f84\u57fa\u5e95\u30ab\u30fc\u30cd\u30eb(RBF Kernel)**\uff1a$k(x, y) = \\exp\\left(-\\frac{\\|x - y\\|^2}{2\\sigma^2}\\right)$\uff08\u7121\u9650\u6b21\u5143\u7279\u5fb4\u7a7a\u9593\u306b\u5bfe\u5fdc\uff09\uff1b\n- **\u30b7\u30b0\u30e2\u30a4\u30c9\u30ab\u30fc\u30cd\u30eb(Sigmoid Kernel)**\uff1a$k(x, y) = \\tanh(\\alpha x^T y + c)$\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u7fa9 12.3 \u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3 (Definition 12.3 Support Vector Machine)\n\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3(SVM: Support Vector Machine)\u306f\u30ab\u30fc\u30cd\u30eb\u6cd5\u306e\u6700\u3082\u53e4\u5178\u7684\u306a\u5fdc\u7528\u3067\u3042\u308b$^{[23]}$\u3002SVM\u306f\u7279\u5fb4\u7a7a\u9593\u306b\u304a\u3044\u3066\u6700\u5927\u30de\u30fc\u30b8\u30f3\u8d85\u5e73\u9762\u3092\u63a2\u3057\u3001\u305d\u306e\u6c7a\u5b9a\u95a2\u6570\u306f\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30c8\u30eb\u3068\u5206\u985e\u5bfe\u8c61\u30b5\u30f3\u30d7\u30eb\u306e\u5185\u7a4d\u306e\u307f\u306b\u4f9d\u5b58\u3059\u308b:\n\n$$f(x) = \\text{sign}\\left(\\sum_{i=1}^{m} \\alpha_i y_i \\langle \\phi(x_i), \\phi(x) \\rangle + b\\right) = \\text{sign}\\left(\\sum_{i=1}^{m} \\alpha_i y_i k(x_i, x) + b\\right) \\tag{12.2}$$\n\n\u3053\u3053\u3067 $x_i$ \u306f\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30c8\u30eb\u3001$y_i \\in \\{-1, +1\\}$ \u306f\u30e9\u30d9\u30eb\u3001$\\alpha_i$ \u306f\u53cc\u5bfe\u5909\u6570\u3067\u3042\u308b\u3002\n```\n\n### 12.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)\n\n\u30ab\u30fc\u30cd\u30eb\u30c8\u30ea\u30c3\u30af\u306e\u5e7e\u4f55\u5b66\u7684\u76f4\u89b3\u306f\u300c\u6298\u308a\u7573\u307f-\u5c55\u958b\u300d\u3068\u3057\u3066\u7406\u89e3\u3067\u304d\u308b:\n\n1. **\u5165\u529b\u7a7a\u9593**\uff1a\u30c7\u30fc\u30bf\u70b9\u306f\u4f4e\u6b21\u5143\u7a7a\u9593\u306b\u4e71\u96d1\u306b\u5206\u5e03\u3057\u3001\u7dda\u5f62\u5206\u985e\u5668\u306f\u5f79\u306b\u7acb\u305f\u306a\u3044\uff1b\n2. **\u6697\u9ed9\u7684\u5199\u50cf $\\phi$**\uff1a\u30c7\u30fc\u30bf\u70b9\u3092\u9ad8\u6b21\u5143\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306b\u300c\u5c55\u958b\u300d\u3057\u3001\u3082\u3064\u308c\u5408\u3063\u3066\u3044\u305f\u30c7\u30fc\u30bf\u70b9\u304c\u300c\u5f15\u304d\u4f38\u3070\u3055\u308c\u308b\u300d\uff1b\n3. **\u9ad8\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u5185\u7a4d**\uff1aSVM\u306f\u9ad8\u6b21\u5143\u7a7a\u9593\u3067\u6700\u5927\u30de\u30fc\u30b8\u30f3\u8d85\u5e73\u9762\u3092\u63a2\u7d22\u3059\u308b\u2014\u2014\u3053\u308c\u306f\u5165\u529b\u7a7a\u9593\u306b\u304a\u3051\u308b\u975e\u7dda\u5f62\u6c7a\u5b9a\u5883\u754c\u3068\u7b49\u4fa1\u3067\u3042\u308b\uff1b\n4. **\u30ab\u30fc\u30cd\u30eb\u95a2\u6570 $k(x, y)$**\uff1a\u30c7\u30fc\u30bf\u304c\u9ad8\u6b21\u5143\u7a7a\u9593\u306b\u5199\u50cf\u3055\u308c\u305f\u304b\u306e\u3088\u3046\u306b\u3001\u9ad8\u6b21\u5143\u7a7a\u9593\u4e2d\u306e\u5185\u7a4d\u5024\u3092\u76f4\u63a5\u8fd4\u3059\u304c\u3001\u8a08\u7b97\u91cf\u306f\u4f4e\u6b21\u5143\u7a7a\u9593\u3068\u540c\u3058\u3067\u3042\u308b\u3002\n\n**\u91cd\u8981\u306a\u6d1e\u5bdf**\uff1aRBF\u30ab\u30fc\u30cd\u30eb $\\exp(-\\gamma\\|x - y\\|^2)$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\u3059\u3079\u3066\u306e\u6b21\u6570\u306e\u591a\u9805\u5f0f\u7279\u5fb4\u3092\u542b\u3080\u305f\u3081\u3001RBF\u30ab\u30fc\u30cd\u30ebSVM\u306f\u7406\u8ad6\u4e0a\u3001\u4efb\u610f\u306e\u8907\u96d1\u306a\u6c7a\u5b9a\u5883\u754c\u3092\u8fd1\u4f3c\u3067\u304d\u308b\u3002\n\n### 12.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u984c 12.1 \u4e8c\u6b21\u5143XOR\u554f\u984c\u306e\u30ab\u30fc\u30cd\u30eb\u30c8\u30ea\u30c3\u30af \u2014 \u624b\u8a08\u7b97\u306b\u3088\u308b\u5c0e\u51fa (Example 12.1 Kernel Trick for 2D XOR Problem \u2014 Manual Derivation)\n\nXOR\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\uff1a$x_1 = (-1, -1)$ \u30e9\u30d9\u30eb $-1$\u3001$x_2 = (1, 1)$ \u30e9\u30d9\u30eb $-1$\u3001$x_3 = (-1, 1)$ \u30e9\u30d9\u30eb $+1$\u3001$x_4 = (1, -1)$ \u30e9\u30d9\u30eb $+1$\u3002\u4e8c\u6b21\u5143\u7a7a\u9593\u3067\u306f\u3001XOR\u30c7\u30fc\u30bf\u306f\u7dda\u5f62\u5206\u96e2\u4e0d\u53ef\u80fd\u3067\u3042\u308b\u3002\n\n**\u89e3**\uff1a\n\n**\u30b9\u30c6\u30c3\u30d7 1\uff1a\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u3092\u9078\u629e\u3057\u3001\u6697\u9ed9\u7684\u5199\u50cf\u3092\u898b\u3064\u3051\u308b\u3002** \u591a\u9805\u5f0f\u30ab\u30fc\u30cd\u30eb $k(x, y) = (x^T y)^2$ \u3092\u53d6\u308b\u3002\u5c55\u958b:\n\n$$(x^T y)^2 = (x_1 y_1 + x_2 y_2)^2 = x_1^2 y_1^2 + 2x_1 x_2 y_1 y_2 + x_2^2 y_2^2$$\n\n$$= \\langle (x_1^2, \\sqrt{2}x_1 x_2, x_2^2), (y_1^2, \\sqrt{2}y_1 y_2, y_2^2) \\rangle$$\n\n\u3057\u305f\u304c\u3063\u3066\u6697\u9ed9\u7684\u5199\u50cf\u306f $\\phi(x) = (x_1^2, \\sqrt{2}x_1 x_2, x_2^2)$ \u3067\u3042\u308a\u3001\u4e8c\u6b21\u5143\u30c7\u30fc\u30bf\u3092\u4e09\u6b21\u5143\u7a7a\u9593\u306b\u5199\u50cf\u3059\u308b\u3002\n\n**\u30b9\u30c6\u30c3\u30d7 2\uff1a\u30c7\u30fc\u30bf\u70b9\u306e\u4e09\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u5ea7\u6a19\u3092\u8a08\u7b97\u3059\u308b\u3002**\n\n$$\\phi(x_1) = \\phi(-1, -1) = (1, \\sqrt{2}, 1), \\quad \\phi(x_2) = \\phi(1, 1) = (1, \\sqrt{2}, 1)$$\n$$\\phi(x_3) = \\phi(-1, 1) = (1, -\\sqrt{2}, 1), \\quad \\phi(x_4) = \\phi(1, -1) = (1, -\\sqrt{2}, 1)$$\n\n**\u30b9\u30c6\u30c3\u30d7 3\uff1a\u7dda\u5f62\u5206\u96e2\u53ef\u80fd\u6027\u3092\u691c\u8a3c\u3059\u308b\u3002** \u4e09\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3044\u3066\u3001$x_1, x_2$\uff08\u30e9\u30d9\u30eb $-1$\uff09\u306f\u3068\u3082\u306b $(1, \\sqrt{2}, 1)$ \u306b\u4f4d\u7f6e\u3057\u3001$x_3, x_4$\uff08\u30e9\u30d9\u30eb $+1$\uff09\u306f\u3068\u3082\u306b $(1, -\\sqrt{2}, 1)$ \u306b\u4f4d\u7f6e\u3059\u308b\u3002\u4e8c\u3064\u306e\u30af\u30e9\u30b9\u306f\u5e73\u9762 $z_2 = 0$\uff08\u3059\u306a\u308f\u3061 $\\sqrt{2}x_1 x_2 = 0$\uff09\u306b\u3088\u3063\u3066\u5b8c\u5168\u306b\u5206\u96e2\u3067\u304d\u308b\uff01\n\n**\u30b9\u30c6\u30c3\u30d7 4\uff1a\u30ab\u30fc\u30cd\u30eb\u30c8\u30ea\u30c3\u30af\u3092\u691c\u8a3c\u3059\u308b\u3002** $k(x_1, x_3) = (x_1^T x_3)^2$ \u3092\u8a08\u7b97:\n\n$$x_1^T x_3 = (-1)(-1) + (-1)(1) = 0, \\quad k(x_1, x_3) = 0^2 = 0$$\n\n\u4e09\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3044\u3066: $\\langle \\phi(x_1), \\phi(x_3) \\rangle = 1 \\times 1 + \\sqrt{2} \\times (-\\sqrt{2}) + 1 \\times 1 = 0$\n\n\u4e21\u8005\u306f\u7b49\u3057\u304f\u3001\u30ab\u30fc\u30cd\u30eb\u30c8\u30ea\u30c3\u30af\u306e\u6b63\u3057\u3055\u304c\u691c\u8a3c\u3055\u308c\u305f\u3002\n\n**\u30b9\u30c6\u30c3\u30d7 5\uff1aSVM\u6c7a\u5b9a\u3002** \u4e09\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3044\u3066\u3001\u6700\u5927\u30de\u30fc\u30b8\u30f3\u8d85\u5e73\u9762\u306f $z_2 = 0$\u3001\u6cd5\u7dda\u30d9\u30af\u30c8\u30eb $w = (0, 1, 0)$\u3001\u30d0\u30a4\u30a2\u30b9 $b = 0$ \u3067\u3042\u308b\u3002\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30c8\u30eb\u306f\u5168\u56db\u70b9\u3001$\\alpha_i = 1$\u3002\n\n\u30c6\u30b9\u30c8\u70b9 $x = (0.5, -0.5)$ \u306b\u3064\u3044\u3066:\n\n$$k(x_1, x) = ((-1)(0.5) + (-1)(-0.5))^2 = 0, \\quad k(x_2, x) = ((1)(0.5) + (1)(-0.5))^2 = 0$$\n$$k(x_3, x) = ((-1)(0.5) + (1)(-0.5))^2 = 1, \\quad k(x_4, x) = ((1)(0.5) + (-1)(-0.5))^2 = 1$$\n\n$$f(x) = \\text{sign}(-0 - 0 + 1 + 1) = \\text{sign}(2) = +1$$\n\n$+1$ \u3068\u4e88\u6e2c\u3055\u308c\u3001\u6b63\u3057\u3044\u3002\n\n**\u91cd\u8981\u306a\u89b3\u5bdf**\uff1a\u6211\u3005\u306f $\\phi(x)$ \u3092\u660e\u793a\u7684\u306b\u8a08\u7b97\u3057\u305f\u3053\u3068\u306f\u4e00\u5ea6\u3082\u306a\u304f\u3001\u30ab\u30fc\u30cd\u30eb\u95a2\u6570 $k(x, y) = (x^T y)^2$ \u3092\u901a\u3058\u3066\u9ad8\u6b21\u5143\u7a7a\u9593\u4e2d\u306e\u5185\u7a4d\u3092\u76f4\u63a5\u5f97\u305f\u2014\u2014\u4f4e\u6b21\u5143\u306e\u8a08\u7b97\u91cf\u3067\u9ad8\u6b21\u5143\u306e\u5206\u985e\u80fd\u529b\u3092\u5b9f\u73fe\u3057\u3066\u3044\u308b\u3002\n```\n\n### 12.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)\n\n\u30ab\u30fc\u30cd\u30eb\u6cd5\u306e\u5fdc\u7528\u306fSVM\u306b\u3068\u3069\u307e\u3089\u306a\u3044:\n\n- **\u30ab\u30fc\u30cd\u30eb\u4e3b\u6210\u5206\u5206\u6790(Kernel PCA)**\uff1a\u30ab\u30fc\u30cd\u30eb\u5199\u50cf\u5f8c\u306e\u9ad8\u6b21\u5143\u7a7a\u9593\u3067PCA\u3092\u884c\u3044\u3001\u975e\u7dda\u5f62\u6b21\u5143\u524a\u6e1b\u3092\u5b9f\u73fe\uff1b\n- **\u30ab\u30fc\u30cd\u30eb\u30ea\u30c3\u30b8\u56de\u5e30(Kernel Ridge Regression)**\uff1a\u7dda\u5f62\u30ea\u30c3\u30b8\u56de\u5e30\u3092\u975e\u7dda\u5f62\u56de\u5e30\u306b\u4e00\u822c\u5316\uff1b\n- **\u30ab\u30fc\u30cd\u30eb\u5e73\u5747\u30de\u30c3\u30c1\u30f3\u30b0(Kernel Mean Matching)**\uff1a\u30c9\u30e1\u30a4\u30f3\u9069\u5fdc\u3068\u8ee2\u79fb\u5b66\u7fd2\u306b\u7528\u3044\u3089\u308c\u308b\uff1b\n- **\u30ac\u30a6\u30b9\u904e\u7a0b(Gaussian Process)**\uff1a\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u3092\u5171\u5206\u6563\u95a2\u6570\u3068\u3057\u3066\u7528\u3044\u3001\u30d9\u30a4\u30ba\u6700\u9069\u5316\u3068\u56de\u5e30\u306b\u5fdc\u7528\uff1b\n- **\u795e\u7d4c\u6b63\u63a5\u30ab\u30fc\u30cd\u30eb(NTK: Neural Tangent Kernel)**\uff1a\u7121\u9650\u5e45\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u30ab\u30fc\u30cd\u30eb\u6cd5\u3092\u7d50\u3073\u3064\u3051\u3001\u6df1\u5c64\u5b66\u7fd2\u306b\u7406\u8ad6\u7684\u5206\u6790\u30c4\u30fc\u30eb\u3092\u63d0\u4f9b\u3002\n\n---\n\n## \u7b2c13\u7ae0 \u91cf\u5b50\u529b\u5b66\u306b\u304a\u3051\u308b\u5185\u7a4d \u2014 \u78ba\u7387\u5373\u5c04\u5f71 (Chapter 13 Inner Products in Quantum Mechanics \u2014 Probability Is Projection)\n\n### 13.1 \u7406\u8ad6\u3068\u53b3\u5bc6\u306a\u5b9a\u7fa9 (Theory and Rigorous Definitions)\n\n\u91cf\u5b50\u529b\u5b66\u306f\u5185\u7a4d\u306e\u6982\u5ff5\u3092\u7269\u7406\u4e16\u754c\u306e\u7a76\u6975\u306e\u30ec\u30d9\u30eb\u3078\u3068\u62bc\u3057\u4e0a\u3052\u308b\u3002\u91cf\u5b50\u529b\u5b66\u306b\u304a\u3044\u3066\u3001\u30b7\u30b9\u30c6\u30e0\u306e\u72b6\u614b\u306f\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593 $\\mathcal{H}$ \u4e0a\u306e**\u72b6\u614b\u30d9\u30af\u30c8\u30eb(State Vector)** $|\\psi\\rangle$ \u306b\u3088\u3063\u3066\u8a18\u8ff0\u3055\u308c\u308b\uff08\u30c7\u30a3\u30e9\u30c3\u30af\u8a18\u6cd5\uff09$^{[26]}$\u3002\u3053\u3053\u3067\u306e\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306f\u901a\u5e38\u3001\u7121\u9650\u6b21\u5143\u306e\u8907\u7d20\u5185\u7a4d\u7a7a\u9593\u3067\u3042\u308b\u3002\n\n```ad-definition\ntitle: \u5b9a\u7fa9 13.1 \u72b6\u614b\u30d9\u30af\u30c8\u30eb\u3068\u5185\u7a4d (Definition 13.1 State Vector and Inner Product)\n\u72b6\u614b\u30d9\u30af\u30c8\u30eb $|\\psi\\rangle \\in \\mathcal{H}$ \u306f\u91cf\u5b50\u30b7\u30b9\u30c6\u30e0\u306e\u3059\u3079\u3066\u306e\u60c5\u5831\u3092\u542b\u3080\u3002\u4e8c\u3064\u306e\u72b6\u614b\u306e\u5185\u7a4d $\\langle \\phi | \\psi \\rangle$ \u306f\u8907\u7d20\u6570\u3067\u3042\u308a\u3001\u305d\u306e\u7d76\u5bfe\u5024\u306e\u4e8c\u4e57\u304c\u6e2c\u5b9a\u78ba\u7387\u3092\u4e0e\u3048\u308b\u3002\n\n**\u516c\u7406 13.1\uff08\u30dc\u30eb\u30f3\u5247(Born Rule)\uff09** \u30b7\u30b9\u30c6\u30e0\u304c\u72b6\u614b $|\\psi\\rangle$ \u306b\u3042\u308b\u3068\u304d\u3001\u53ef\u89b3\u6e2c\u91cf $\\hat{A}$ \u3092\u6e2c\u5b9a\u3057\u3066\u56fa\u6709\u5024 $\\lambda_n$ \u3092\u5f97\u308b\u78ba\u7387\u306f$^{[21]}$:\n\n$$P(\\lambda_n) = |\\langle a_n | \\psi \\rangle|^2 \\tag{13.1}$$\n\n\u3053\u3053\u3067 $|a_n\\rangle$ \u306f $\\hat{A}$ \u306e $\\lambda_n$ \u306b\u5bfe\u5fdc\u3059\u308b\u56fa\u6709\u72b6\u614b\u3067\u3042\u308b\u3002\u6e2c\u5b9a\u5f8c\u3001\u30b7\u30b9\u30c6\u30e0\u306e\u72b6\u614b\u306f $|a_n\\rangle$ \u306b\u53ce\u7e2e\u3059\u308b\u3002\u30dc\u30eb\u30f3\u5247\u306e\u672c\u8cea\u306f\uff1a**\u78ba\u7387\u306f\u6e2c\u5b9a\u57fa\u5e95\u306b\u304a\u3051\u308b\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306e\u5c04\u5f71\u306e\u7d76\u5bfe\u5024\u4e8c\u4e57\u306b\u7b49\u3057\u3044**\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u7fa9 13.2 \u53ef\u89b3\u6e2c\u91cf\u3068\u81ea\u5df1\u5171\u5f79\u6f14\u7b97\u5b50 (Definition 13.2 Observables and Self-Adjoint Operators)\n\u53ef\u89b3\u6e2c\u91cf\u306f\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u4e0a\u306e\u81ea\u5df1\u5171\u5f79\u6f14\u7b97\u5b50(Hermitian Operator) $\\hat{A}$ \u306b\u5bfe\u5fdc\u3057\u3001$\\hat{A}^\\dagger = \\hat{A}$ \u3092\u6e80\u305f\u3059\u3002\u81ea\u5df1\u5171\u5f79\u6f14\u7b97\u5b50\u306e\u56fa\u6709\u5024\u306f\u5b9f\u6570\u3067\u3042\u308a\u3001\u56fa\u6709\u72b6\u614b\u306f\u5b8c\u5099\u306a\u76f4\u4ea4\u57fa\u5e95\u3092\u69cb\u6210\u3059\u308b\u3002\n```\n\n```ad-definition\ntitle: \u5b9a\u7fa9 13.3 \u30b7\u30e5\u30ec\u30fc\u30c7\u30a3\u30f3\u30ac\u30fc\u65b9\u7a0b\u5f0f (Definition 13.3 Schrodinger Equation)\n\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306e\u6642\u9593\u767a\u5c55\u306f\u30b7\u30e5\u30ec\u30fc\u30c7\u30a3\u30f3\u30ac\u30fc\u65b9\u7a0b\u5f0f\u306b\u3088\u3063\u3066\u8a18\u8ff0\u3055\u308c\u308b:\n\n$$i\\hbar \\frac{d}{dt} |\\psi(t)\\rangle = \\hat{H} |\\psi(t)\\rangle \\tag{13.2}$$\n\n\u3053\u3053\u3067 $\\hat{H}$ \u306f\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u6f14\u7b97\u5b50\uff08\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50\uff09\u3067\u3042\u308b\u3002\u3053\u306e\u65b9\u7a0b\u5f0f\u306f\u672c\u8cea\u7684\u306b\u3001\u7121\u9650\u6b21\u5143\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306b\u304a\u3051\u308b\u30e6\u30cb\u30bf\u30ea\u767a\u5c55\u65b9\u7a0b\u5f0f\u2014\u2014\u5185\u7a4d\u3092\u4fdd\u5b58\u3059\u308b\u56de\u8ee2\u2014\u2014\u3067\u3042\u308b\u3002\n```\n\n### 13.2 \u5e7e\u4f55\u3068\u7a7a\u9593\u30a4\u30e1\u30fc\u30b8 (Geometry and Spatial Intuition)\n\n\u91cf\u5b50\u529b\u5b66\u306e\u5e7e\u4f55\u5b66\u7684\u30a4\u30e1\u30fc\u30b8\u306f\u53e4\u5178\u7684\u306a\u5185\u7a4d\u7a7a\u9593\u3068\u6df1\u3044\u95a2\u9023\u3092\u6301\u3064:\n\n1. **\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306f\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b**\uff1a\u7269\u7406\u7684\u306b\u306f $|\\psi\\rangle$ \u306e\u6b63\u898f\u5316\u304c\u8981\u6c42\u3055\u308c\u3001\u3059\u306a\u308f\u3061 $\\langle \\psi | \\psi \\rangle = 1$\u3002\u3059\u3079\u3066\u306e\u53ef\u80fd\u306a\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306f\u8907\u7d20\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u4e2d\u306e\u5358\u4f4d\u7403\u9762\u3092\u69cb\u6210\u3059\u308b\u3002\n\n2. **\u6e2c\u5b9a\u306f\u76f4\u4ea4\u5c04\u5f71\u3067\u3042\u308b**\uff1a\u6e2c\u5b9a\u64cd\u4f5c\u306f\u72b6\u614b\u30d9\u30af\u30c8\u30eb $|\\psi\\rangle$ \u3092\u56fa\u6709\u90e8\u5206\u7a7a\u9593\u306b\u5c04\u5f71\u3059\u308b\u3002\u5c04\u5f71\u9577 $|\\langle a_n | \\psi \\rangle|$ \u304c\u78ba\u7387\u632f\u5e45\u3092\u6c7a\u5b9a\u3057\u3001\u305d\u306e\u4e8c\u4e57\u304c\u6e2c\u5b9a\u78ba\u7387\u3068\u306a\u308b\u3002\n\n3. **\u76f4\u4ea4\u72b6\u614b\u306f\u4e92\u3044\u306b\u6392\u4ed6\u7684\u3067\u3042\u308b**\uff1a$\\langle \\phi | \\psi \\rangle = 0$ \u306a\u3089\u3070\u3001\u4e8c\u3064\u306e\u72b6\u614b\u306f\u76f4\u4ea4\uff08\u4e92\u3044\u306b\u6392\u4ed6\u7684\uff09\u2014\u2014\u30b7\u30b9\u30c6\u30e0\u304c $|\\psi\\rangle$ \u306b\u3042\u308b\u3068\u304d\u3001$|\\phi\\rangle$ \u304c\u6e2c\u5b9a\u3055\u308c\u308b\u78ba\u7387\u306f\u30bc\u30ed\u3067\u3042\u308b\u3002\n\n4. **\u7d61\u307f\u5408\u3044\u72b6\u614b\u306f\u5206\u96e2\u4e0d\u53ef\u80fd\u3067\u3042\u308b**\uff1a\u8907\u5408\u30b7\u30b9\u30c6\u30e0\u306b\u304a\u3044\u3066\u3001$|\\psi\\rangle_{AB} \\neq |\\phi\\rangle_A \\otimes |\\chi\\rangle_B$ \u306a\u3089\u3070\u3001\u4e8c\u3064\u306e\u90e8\u5206\u30b7\u30b9\u30c6\u30e0\u306f\u7d61\u307f\u5408\u3044\u72b6\u614b\u306b\u3042\u308b\u3002\u7d61\u307f\u5408\u3044\u72b6\u614b\u306e\u6570\u5b66\u7684\u672c\u8cea\u306f\u3001\u4e8c\u3064\u306e\u90e8\u5206\u30b7\u30b9\u30c6\u30e0\u306e\u5185\u7a4d\u69cb\u9020\u304c\u76f4\u7a4d\u5f62\u5f0f\u306b\u5206\u89e3\u3067\u304d\u306a\u3044\u3053\u3068\u3067\u3042\u308b\u3002\n\n### 13.3 \u786c\u6838\u4f8b\u984c\u8a73\u89e3 (Worked Example)\n\n```ad-example\ntitle: \u4f8b\u984c 13.1 \u30b9\u30d4\u30f3 $1\/2$ \u30b7\u30b9\u30c6\u30e0\u306e\u6e2c\u5b9a\u78ba\u7387 \u2014 \u5185\u7a4d\u8a08\u7b97 (Example 13.1 Measurement Probability for a Spin $1\/2$ System \u2014 Inner Product Calculation)\n\n\u96fb\u5b50\u30b9\u30d4\u30f3\u3092\u8003\u3048\u308b\u3002\u305d\u306e\u72b6\u614b\u306f\u4e8c\u6b21\u5143\u8907\u7d20\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u4e2d\u306e\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u8868\u73fe\u3067\u304d\u308b\u3002\u30b9\u30d4\u30f3 $z$ \u65b9\u5411\u306e\u56fa\u6709\u72b6\u614b:\n\n$$| \\uparrow_z \\rangle = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}, \\quad | \\downarrow_z \\rangle = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}$$\n\n\u30b9\u30d4\u30f3 $x$ \u65b9\u5411\u306e\u56fa\u6709\u72b6\u614b:\n\n$$| \\uparrow_x \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}, \\quad | \\downarrow_x \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}$$\n\n\u96fb\u5b50\u306f\u72b6\u614b $|\\psi\\rangle = \\frac{1}{\\sqrt{2}}| \\uparrow_z \\rangle + \\frac{1}{\\sqrt{2}}| \\downarrow_z \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}$ \u306b\u3042\u308b\u3002\n\n**\u89e3**\uff1a\n\n**\u30b9\u30c6\u30c3\u30d7 1\uff1a\u6b63\u898f\u5316\u306e\u691c\u8a3c\u3002**\n\n$$\\langle \\psi | \\psi \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & 1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = \\frac{1}{2}(1 + 1) = 1$$\n\n\u6b63\u898f\u5316\u304c\u6210\u7acb\u3059\u308b\u3002\n\n**\u30b9\u30c6\u30c3\u30d7 2\uff1a$S_z$ \u306e\u6e2c\u5b9a\u78ba\u7387\u3002**\n\n$$P(\\uparrow_z) = |\\langle \\uparrow_z | \\psi \\rangle|^2 = \\left| \\begin{pmatrix} 1 & 0 \\end{pmatrix} \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{\\sqrt{2}} \\right|^2 = \\frac{1}{2}$$\n\n$$P(\\downarrow_z) = |\\langle \\downarrow_z | \\psi \\rangle|^2 = \\left| \\begin{pmatrix} 0 & 1 \\end{pmatrix} \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{\\sqrt{2}} \\right|^2 = \\frac{1}{2}$$\n\n\u540450%\u3067\u3042\u308a\u3001\u671f\u5f85\u901a\u308a\u3067\u3042\u308b\u3002\n\n**\u30b9\u30c6\u30c3\u30d7 3\uff1a$S_x$ \u306e\u6e2c\u5b9a\u78ba\u7387\u3002**\n\n$$P(\\uparrow_x) = |\\langle \\uparrow_x | \\psi \\rangle|^2 = \\left| \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & 1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{2}(1 + 1) \\right|^2 = 1$$\n\n$$P(\\downarrow_x) = |\\langle \\downarrow_x | \\psi \\rangle|^2 = \\left| \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & -1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{2}(1 - 1) \\right|^2 = 0$$\n\n**\u91cd\u8981\u306a\u89b3\u5bdf**\uff1a$|\\psi\\rangle = | \\uparrow_x \\rangle$ \u3067\u3042\u308b\u305f\u3081\u3001$S_x$ \u3092\u6e2c\u5b9a\u3059\u308b\u3068100%\u306e\u78ba\u7387\u3067 $+\\hbar\/2$ \u3092\u5f97\u308b\u3002\u3053\u308c\u306f\u5185\u7a4d\u306e\u5e7e\u4f55\u5b66\u7684\u610f\u5473\u3092\u691c\u8a3c\u3059\u308b\uff1a\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u304c\u5b8c\u5168\u306b\u4e00\u81f4\u3059\u308b\u3068\u304d\uff08\u5185\u7a4d\u306e\u7d76\u5bfe\u5024\u304c1\uff09\u3001\u78ba\u7387\u306f100%\uff1b\u76f4\u4ea4\u3059\u308b\u3068\u304d\uff08\u5185\u7a4d\u304c0\uff09\u3001\u78ba\u7387\u306f0\u3067\u3042\u308b\u3002\n\n**\u30b9\u30c6\u30c3\u30d7 4\uff1a\u6e2c\u5b9a\u5f8c\u306e\u72b6\u614b\u53ce\u7e2e\u3002** $S_z$ \u3092\u6e2c\u5b9a\u3057\u3066 $+\\hbar\/2$ \u3092\u5f97\u305f\u3068\u4eee\u5b9a\u3059\u308b\u3068\u3001\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306f\u53ce\u7e2e\u3059\u308b:\n\n$$|\\psi\\rangle = \\frac{1}{\\sqrt{2}}| \\uparrow_z \\rangle + \\frac{1}{\\sqrt{2}}| \\downarrow_z \\rangle \\xrightarrow{\\text{\u6e2c\u5b9a } S_z = +\\hbar\/2} |\\psi'\\rangle = | \\uparrow_z \\rangle$$\n\n\u3053\u306e\u3068\u304d\u518d\u5ea6 $S_z$ \u3092\u6e2c\u5b9a\u3059\u308b\u3068100% $+\\hbar\/2$ \u3092\u5f97\u308b\u304c\u3001$S_x$ \u3092\u6e2c\u5b9a\u3059\u308b\u3068\u518d\u307350\/50\u306e\u78ba\u7387\u306b\u623b\u308b\u3002\u3053\u308c\u304c\u300c\u6e2c\u5b9a\u304c\u72b6\u614b\u3092\u5909\u3048\u308b\u300d\u3068\u3044\u3046\u672c\u8cea\u2014\u2014\u76f4\u4ea4\u5c04\u5f71\u64cd\u4f5c\u2014\u2014\u3067\u3042\u308b\u3002\n```\n\n### 13.4 \u5de5\u5b66\u3068\u6700\u5148\u7aef\u5fdc\u7528 (Engineering and Cutting-Edge Applications)\n\n\u91cf\u5b50\u5185\u7a4d\u306e\u6982\u5ff5\u306f\u9769\u547d\u7684\u306a\u6280\u8853\u3092\u751f\u307f\u51fa\u3057\u3064\u3064\u3042\u308b:\n\n- **\u91cf\u5b50\u8a08\u7b97(Quantum Computing)**\uff1a\u91cf\u5b50\u30b2\u30fc\u30c8\u64cd\u4f5c\u306f\u672c\u8cea\u7684\u306b\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306b\u304a\u3051\u308b\u30e6\u30cb\u30bf\u30ea\u5909\u63db\uff08\u5185\u7a4d\u3092\u4fdd\u5b58\u3059\u308b\u56de\u8ee2\uff09\u3067\u3042\u308b\u3002Shor\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068Grover\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u91cf\u5b50\u72b6\u614b\u306e\u91cd\u306d\u5408\u308f\u305b\u3068\u5e72\u6e09\uff08\u5185\u7a4d\u306e\u4f4d\u76f8\uff09\u3092\u5229\u7528\u3057\u3066\u6307\u6570\u95a2\u6570\u7684\u306a\u9ad8\u901f\u5316\u3092\u5b9f\u73fe\u3059\u308b\uff1b\n- **\u91cf\u5b50\u6697\u53f7(Quantum Cryptography)**\uff1aBB84\u30d7\u30ed\u30c8\u30b3\u30eb\u306f\u6e2c\u5b9a\u57fa\u5e95\u306e\u76f4\u4ea4\u6027\u3092\u5229\u7528\u3057\u3066\u76d7\u8074\u3092\u691c\u51fa\u3059\u308b\u2014\u2014\u76d7\u8074\u8005\u306e\u6e2c\u5b9a\u306f\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u3092\u53ce\u7e2e\u3055\u305b\u3001\u5185\u7a4d\u7d50\u679c\u3092\u5909\u5316\u3055\u305b\u308b\u305f\u3081\u3001\u6b63\u5f53\u306a\u901a\u4fe1\u8005\u304c\u767a\u898b\u3067\u304d\u308b\uff1b\n- **\u91cf\u5b50\u30c6\u30ec\u30dd\u30fc\u30c6\u30fc\u30b7\u30e7\u30f3(Quantum Teleportation)**\uff1aBell\u72b6\u614b\uff08\u6700\u5927\u7d61\u307f\u5408\u3044\u72b6\u614b\uff09\u306e\u5185\u7a4d\u69cb\u9020\u3092\u5229\u7528\u3057\u3066\u91cf\u5b50\u60c5\u5831\u306e\u9060\u9694\u8ee2\u9001\u3092\u5b9f\u73fe\u3059\u308b\uff1b\n- **\u91cf\u5b50\u6a5f\u68b0\u5b66\u7fd2(Quantum Machine Learning)**\uff1a\u91cf\u5b50\u30ab\u30fc\u30cd\u30eb\u6cd5\u306f\u91cf\u5b50\u72b6\u614b\u306e\u5185\u7a4d\u3092\u5229\u7528\u3057\u3066\u9ad8\u6b21\u5143\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u3067\u52b9\u7387\u7684\u306b\u30ab\u30fc\u30cd\u30eb\u95a2\u6570\u3092\u8a08\u7b97\u3057\u3001\u91cf\u5b50\u512a\u4f4d\u6027\u306e\u5b9f\u73fe\u304c\u671f\u5f85\u3055\u308c\u308b\u3002\n\n---\n\n## \u7d42\u7ae0 \u5927\u7d71\u4e00\u77e5\u8b58\u30b0\u30e9\u30d5\u3068\u54f2\u5b66\u7684\u6607\u83ef (Final Chapter Grand Unified Knowledge Graph and Philosophical Sublimation)\n\n### \u4e07\u7269\u306f\u5c04\u5f71\u3067\u3042\u308b\u2014\u2014\u5168\u5b66\u554f\u3092\u8cab\u304f\u5185\u7a4d\u306e\u30b0\u30e9\u30d5 (Everything Is a Projection \u2014 An Inner Product Map Across All Disciplines)\n\n\u672c\u8ad6\u6587\u3067\u69cb\u7bc9\u3057\u305f\u77e5\u8b58\u4f53\u7cfb\u3092\u632f\u308a\u8fd4\u308b\u3068\u3001\u4e8c\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u30c9\u30c3\u30c8\u7a4d\u304b\u3089\u7121\u9650\u6b21\u5143\u8907\u7d20\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593\u306e\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u5185\u7a4d\u306b\u81f3\u308b\u307e\u3067\u3001\u5185\u7a4d\u306e\u6982\u5ff5\u306f\u6570\u5b66\u3001\u7269\u7406\u5b66\u3001\u5de5\u5b66\u3001\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u79d1\u5b66\u306e\u3042\u3089\u3086\u308b\u9685\u3005\u3092\u8cab\u3044\u3066\u3044\u308b\u3002\n\n**\u6838\u5fc3\u7684\u4e3b\u7dda**\uff1a\u5185\u7a4d $\\langle \\cdot, \\cdot \\rangle$ \u306f**\u985e\u4f3c\u5ea6\u5c3a\u5ea6(Similarity Measure)** \u3067\u3042\u308b\u3002\u5bfe\u8c61\u304c\u30d9\u30af\u30c8\u30eb\u3001\u95a2\u6570\u3001\u4fe1\u53f7\u3001\u753b\u50cf\u3001\u91cf\u5b50\u72b6\u614b\u306e\u3044\u305a\u308c\u3067\u3042\u3063\u3066\u3082\u3001\u5185\u7a4d\u306f\u540c\u3058\u554f\u3044\u306b\u7b54\u3048\u308b\u2014\u2014\u300c\u3053\u306e\u4e8c\u3064\u306e\u5bfe\u8c61\u306f\u3069\u306e\u7a0b\u5ea6\u4f3c\u3066\u3044\u308b\u304b\uff1f\u300d\n\n**\u5927\u7d71\u4e00\u77e5\u8b58\u30b0\u30e9\u30d5(Grand Unified Knowledge Graph)**:\n\n| \u5206\u91ce | \u5185\u7a4d\u306e\u5177\u4f53\u7684\u306a\u5f62\u5f0f | \u5e7e\u4f55\u5b66\u7684\u89e3\u91c8 | \u4e2d\u6838\u7684\u5fdc\u7528 |\n|------|--------------|---------|---------|\n| \u7dda\u5f62\u4ee3\u6570 | $\\langle x, y \\rangle = x^T y$ | \u5c04\u5f71\u9577 | \u76f4\u4ea4\u5206\u89e3\u3001\u6700\u5c0f\u4e8c\u4e57 |\n| \u95a2\u6570\u89e3\u6790 | $\\langle f, g \\rangle = \\int fg$ | \u6ce2\u5f62\u985e\u4f3c\u5ea6 | \u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3001\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db |\n| \u4fe1\u53f7\u51e6\u7406 | $\\langle x, h \\rangle = \\sum x[n]h[n]$ | \u30de\u30c3\u30c1\u30c9\u30d5\u30a3\u30eb\u30bf | \u7573\u307f\u8fbc\u307f\u3001\u76f8\u95a2\u691c\u51fa |\n| \u78ba\u7387\u7d71\u8a08 | $\\text{Cov}(X,Y) = E[(X-\\mu_X)(Y-\\mu_Y)]$ | \u76f8\u95a2\u65b9\u5411 | PCA\u3001\u56de\u5e30\u5206\u6790 |\n| \u6a5f\u68b0\u5b66\u7fd2 | $\\langle Q_i, K_j \\rangle$ | \u6ce8\u610f\u91cd\u307f | Transformer\u3001\u81ea\u5df1\u6ce8\u610f |\n| \u753b\u50cf\u51e6\u7406 | $\\langle I, K \\rangle$ | \u7279\u5fb4\u5fdc\u7b54 | \u7573\u307f\u8fbc\u307f\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3001\u30a8\u30c3\u30b8\u691c\u51fa |\n| \u91cf\u5b50\u529b\u5b66 | $\\langle \\phi \\mid \\psi \\rangle$ | \u78ba\u7387\u632f\u5e45 | \u6e2c\u5b9a\u3001\u91cf\u5b50\u8a08\u7b97 |\n| \u5236\u5fa1\u7406\u8ad6 | $\\langle f, e^{-st} \\rangle$ | \u8907\u7d20\u5468\u6ce2\n\u6570\u9818\u57df\u5c04\u5f71 | \u30e9\u30d7\u30e9\u30b9\u5909\u63db\u3001\u5b89\u5b9a\u6027\u89e3\u6790 |\n\n### \u54f2\u5b66\u7684\u6607\u83ef\u2014\u2014\u5c04\u5f71\u5373\u8a8d\u77e5 (Philosophical Sublimation \u2014 Projection Is Cognition)\n\n\u54f2\u5b66\u306e\u30ec\u30d9\u30eb\u304b\u3089\u898b\u308b\u3068\u3001\u300c\u4e07\u7269\u306f\u5c04\u5f71\u3067\u3042\u308b\u300d\u306f\u6570\u5b66\u7684\u547d\u984c\u3067\u3042\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u4e16\u754c\u3092\u8a8d\u8b58\u3059\u308b\u4e00\u3064\u306e\u65b9\u6cd5\u3067\u3082\u3042\u308b$^{[22]}$:\n\n1. **\u8a8d\u77e5\u5373\u5c04\u5f71**\uff1a\u4eba\u9593\u304c\u4e16\u754c\u3092\u8a8d\u8b58\u3059\u308b\u30d7\u30ed\u30bb\u30b9\u306f\u3001\u672c\u8cea\u7684\u306b\u5916\u90e8\u4e16\u754c\u306e\u8907\u96d1\u306a\u60c5\u5831\u3092\u6709\u9650\u306e\u8a8d\u77e5\u57fa\u5e95\u95a2\u6570\u306b\u5c04\u5f71\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002\u6211\u3005\u304c\u898b\u3066\u3044\u308b\u306e\u306f\u300c\u73fe\u5b9f\u4e16\u754c\u305d\u306e\u3082\u306e\u300d\u3067\u306f\u306a\u304f\u3001\u8a8d\u77e5\u57fa\u5e95\u4e0a\u3067\u306e\u73fe\u5b9f\u4e16\u754c\u306e\u5c04\u5f71\u4fc2\u6570\u3067\u3042\u308b\u3002\n\n2. **\u76f4\u4ea4\u5373\u72ec\u7acb**\uff1a\u4e8c\u3064\u306e\u6982\u5ff5\u304c\u76f4\u4ea4\u3059\u308b\u3068\u304d\u3001\u305d\u308c\u3089\u306f\u4e92\u3044\u306b\u5e72\u6e09\u305b\u305a\u3001\u91cd\u306a\u308a\u5408\u308f\u306a\u3044\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u76f4\u4ea4\u5206\u89e3\u306f\u8907\u96d1\u306a\u554f\u984c\u3092\u5358\u7d14\u5316\u3059\u308b\u7a76\u6975\u306e\u6b66\u5668\u3067\u3042\u308b\u2014\u2014\u8907\u96d1\u306a\u30b7\u30b9\u30c6\u30e0\u3092\u4e92\u3044\u306b\u76f8\u95a2\u306e\u306a\u3044\u72ec\u7acb\u3057\u305f\u30e2\u30b8\u30e5\u30fc\u30eb\u306b\u5206\u89e3\u3059\u308b\u3002\n\n3. **\u5c04\u5f71\u5373\u6c7a\u5b9a**\uff1a\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306f\u3001\u6b63\u78ba\u306a\u89e3\u304c\u5b58\u5728\u3057\u306a\u3044\u3068\u304d\u306b\u5c04\u5f71\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u6700\u9069\u306a\u9078\u629e\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u5b8c\u5168\u306a\u89e3\u6c7a\u7b56\u304c\u5f97\u3089\u308c\u306a\u3044\u3068\u304d\u3001\u5b9f\u884c\u53ef\u80fd\u9818\u57df\u4e0a\u306b\u76f4\u4ea4\u5c04\u5f71\u3092\u884c\u3046\u3053\u3068\u304c\u6700\u9069\u306a\u6c7a\u5b9a\u3068\u306a\u308b\u3002\n\n4. **\u57fa\u5e95\u306e\u9078\u629e\u304c\u3059\u3079\u3066\u3092\u6c7a\u5b9a\u3059\u308b**\uff1a\u30d5\u30fc\u30ea\u30a8\u306f\u6b63\u5f26\u6ce2\u3092\u57fa\u5e95\u306b\u9078\u3073\u3001\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u30b5\u30dd\u30fc\u30c8\u95a2\u6570\u3092\u57fa\u5e95\u306b\u9078\u3073\u3001Transformer\u306f\u5b66\u7fd2\u53ef\u80fd\u306a\u6ce8\u610f\u57fa\u5e95\u3092\u9078\u3076\u2014\u2014\u3069\u306e\u3088\u3046\u306a\u57fa\u5e95\u3092\u9078\u3076\u304b\u304c\u3001\u3069\u306e\u3088\u3046\u306a\u4e16\u754c\u304c\u898b\u3048\u308b\u304b\u3092\u6c7a\u5b9a\u3059\u308b\u3002\n\n### \u7d42\u5c40\u306e\u601d\u8003 (Final Thoughts)\n\n\u5185\u7a4d\u306f\u5358\u306a\u308b\u6570\u5b66\u7684\u6f14\u7b97\u3067\u306f\u306a\u304f\u3001\u30df\u30af\u30ed\u3068\u30de\u30af\u30ed\u3001\u9023\u7d9a\u3068\u96e2\u6563\u3001\u6c7a\u5b9a\u8ad6\u3068\u78ba\u7387\u8ad6\u3092\u7d50\u3073\u3064\u3051\u308b**\u30e1\u30bf\u8a00\u8a9e(Meta-Language)** \u3067\u3042\u308b\u3002\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u304b\u3089\u91cf\u5b50\u3082\u3064\u308c\u307e\u3067\u3001\u6700\u5c0f\u4e8c\u4e57\u6cd5\u304b\u3089\u5927\u898f\u6a21\u8a00\u8a9e\u30e2\u30c7\u30eb\u307e\u3067\u3001\u5185\u7a4d\u306f\u305d\u306e\u7c21\u6f54\u304b\u3064\u6df1\u9060\u306a\u5f62\u5f0f\u306b\u3088\u3063\u3066\u3001\u4eba\u985e\u306e\u77e5\u8b58\u4f53\u7cfb\u306e\u3042\u3089\u3086\u308b\u9818\u57df\u3092\u7d71\u4e00\u3057\u3066\u3044\u308b\u3002\n\n---\n\n## \u4ed8\u9332 \u672c\u7a3f\u306e\u56f3\u8868\u751f\u6210\u30b3\u30fc\u30c9 (Appendix Code for Generating Figures in This Paper)\n\n\u672c\u7a3f\u306e\u5168\u4e94\u679a\u306e\u56f3\u8868\uff08\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u30d2\u30fc\u30c8\u30de\u30c3\u30d7\u3001\u6700\u5c0f\u4e8c\u4e57\u5c04\u5f71\u3001\u30d5\u30fc\u30ea\u30a8\u5206\u89e3\u3001\u7573\u307f\u8fbc\u307f\u30de\u30c3\u30c1\u30c9\u30d5\u30a3\u30eb\u30bf\u3001Sobel\u30a8\u30c3\u30b8\u691c\u51fa\uff09\u306f\u3001main.py<\/a> \u306b\u3088\u3063\u3066\u4e00\u5143\u7684\u306b\u751f\u6210\u3055\u308c\u3066\u3044\u308b\u3002\u3053\u306e\u30b9\u30af\u30ea\u30d7\u30c8\u306fPython\u306e\u79d1\u5b66\u8a08\u7b97\u30a8\u30b3\u30b7\u30b9\u30c6\u30e0\uff08NumPy\u3001SciPy\u3001Matplotlib\uff09\u306b\u57fa\u3065\u304d\u3001\u300c\u5185\u7a4d\u300d\u3068\u3044\u3046\u6838\u5fc3\u30c6\u30fc\u30de\u3092\u4e2d\u5fc3\u306b\u3001\u6587\u4e2d\u306e\u62bd\u8c61\u7684\u306a\u6570\u5b66\u6982\u5ff5\u3092\u76f4\u611f\u7684\u306a\u53ef\u8996\u5316\u30b0\u30e9\u30d5\u30a3\u30c3\u30af\u306b\u5909\u63db\u3057\u3066\u3044\u308b\u3002<\/p>\n \u30b9\u30af\u30ea\u30d7\u30c8\u306e\u6838\u5fc3\u7684\u306a\u8a2d\u8a08\u601d\u60f3\u306f\u4ee5\u4e0b\u306e\u901a\u308a:<\/p>\n 1. **\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6**\uff1a`cosine_similarity()` \u95a2\u6570\u306b\u3088\u308a\u5358\u8a9e\u57cb\u3081\u8fbc\u307f\u30d9\u30af\u30c8\u30eb\u9593\u306e\u6b63\u898f\u5316\u5185\u7a4d\u3092\u8a08\u7b97\u3057\u3001$5 \\times 5$ \u306e\u30d2\u30fc\u30c8\u30de\u30c3\u30d7\u884c\u5217\u3092\u751f\u6210\u3059\u308b\u3002\u3053\u306e\u95a2\u6570\u306f\u5f0f (1.5) \u306e\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u5b9a\u7fa9\u3092\u5b9f\u88c5\u3057\u3066\u3044\u308b\u3002 \u4ee5\u4e0b\u306f\u30b9\u30af\u30ea\u30d7\u30c8\u5185\u3067\u30b3\u30b5\u30a4\u30f3\u985e\u4f3c\u5ea6\u30d2\u30fc\u30c8\u30de\u30c3\u30d7\u3092\u751f\u6210\u3059\u308b\u4e2d\u6838\u7684\u306a\u30b3\u30fc\u30c9\u65ad\u7247\u3067\u3042\u308b:<\/p>\n \u5b8c\u5168\u306a\u30b3\u30fc\u30c9\u306f main.py<\/a> \u3092\u53c2\u7167\u3055\u308c\u305f\u3044\u3002<\/p>\n ## \u53c2\u8003\u6587\u732e (References)<\/p>\n [1] Wikipedia contributors. (2026, April 28). Dot product. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:42, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Dot_product&oldid=1351567929<\/a>.<\/p>\n [2] Wikipedia contributors. (2025, November 3). Orthogonal complement. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:43, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Orthogonal_complement&oldid=1320174088<\/a>.<\/p>\n [3] Wikipedia contributors. (2025, July 7). Orthogonalization. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:44, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Orthogonalization&oldid=1299273509<\/a>.<\/p>\n [4] Wikipedia contributors. (2025, September 1). Orthogonal functions. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:46, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Orthogonal_functions&oldid=1308940353<\/a>.<\/p>\n [5] Wikipedia contributors. (2026, March 13). Least squares. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:46, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Least_squares&oldid=1343263636<\/a>.<\/p>\n [6] Wikipedia contributors. (2026, May 23). Hilbert space. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:47, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Hilbert_space&oldid=1355759876<\/a>.<\/p>\n [7] \u5377\u79ef\u3001\u5185\u79ef\u3001\u4e92\u76f8\u5173\u6982\u5ff5. CSDN\u535a\u5ba2, 2024. https:\/\/blog.csdn.net\/qq_31073871\/article\/details\/146475191<\/a>.<\/p>\n [8] Wikipedia contributors. (2026, February 27). Inner product space. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:51, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Inner_product_space&oldid=1340828148<\/a>.<\/p>\n [9] \u5185\u79ef\u548c\u5916\u79ef[G\/OL]. OI Wiki, 2025. https:\/\/oi-wiki.org\/math\/linear-algebra\/product\/<\/a>.<\/p>\n [10] \u7ef4\u57fa\u767e\u79d1\u7f16\u8005. \u5185\u79ef[G\/OL]. \u7ef4\u57fa\u767e\u79d1, 2025(20250703)[2025-07-03]. https:\/\/zh.wikipedia.org\/w\/index.php?title=%E5%86%85%E7%A7%AF&oldid=88045564<\/a>.<\/p>\n [11] Wikipedia contributors. (2026, April 24). Fourier series. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:55, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Fourier_series&oldid=1350934101<\/a>.<\/p>\n [12] Wikipedia contributors. (2026, May 20). Fourier transform. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:55, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Fourier_transform&oldid=1355147665<\/a>.<\/p>\n [13] Wikipedia contributors. (2026, May 17). Cosine similarity. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:56, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Cosine_similarity&oldid=1354643579<\/a>.<\/p>\n [14] Wikipedia contributors. (2026, May 11). Laplace transform. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:56, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform&oldid=1353668445<\/a>.<\/p>\n [15] Wikipedia contributors. (2026, May 8). Z-transform. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:57, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Z-transform&oldid=1353129057<\/a>.<\/p>\n [16] Wikipedia contributors. (2025, June 1). Frequency domain. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:57, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Frequency_domain&oldid=1293464779<\/a>.<\/p>\n [17] Wikipedia contributors. (2026, May 20). Convolution. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:57, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Convolution&oldid=1355143781<\/a>.<\/p>\n [18] Wikipedia contributors. (2026, April 25). Discrete cosine transform. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:58, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Discrete_cosine_transform&oldid=1350947997<\/a>.<\/p>\n [19] Wikipedia contributors. (2026, May 19). JPEG. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:58, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=JPEG&oldid=1355030069<\/a>.<\/p>\n [20] Wikipedia contributors. (2026, April 29). Wavelet. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:58, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Wavelet&oldid=1351640900<\/a>.<\/p>\n [21] Wikipedia contributors. (2026, March 22). Word embedding. In _Wikipedia, The Free Encyclopedia_. Retrieved 11:59, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Word_embedding&oldid=1344811356<\/a>.<\/p>\n [22] Wikipedia contributors. (2025, November 24). Kernel method. In _Wikipedia, The Free Encyclopedia_. Retrieved 12:00, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Kernel_method&oldid=1323912764<\/a>.<\/p>\n [23] Wikipedia contributors. (2026, April 19). Support vector machine. In _Wikipedia, The Free Encyclopedia_. Retrieved 12:00, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Support_vector_machine&oldid=1350010737<\/a>.<\/p>\n [24] Wikipedia contributors. (2026, May 23). Cluster analysis. In _Wikipedia, The Free Encyclopedia_. Retrieved 12:00, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Cluster_analysis&oldid=1355672094<\/a>.<\/p>\n [25] Wikipedia contributors. (2026, April 8). Regression analysis. In _Wikipedia, The Free Encyclopedia_. Retrieved 12:01, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Regression_analysis&oldid=1347668389<\/a>.<\/p>\n [26] Wikipedia contributors. (2026, May 22). Quantum mechanics. In _Wikipedia, The Free Encyclopedia_. Retrieved 12:01, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_mechanics&oldid=1355584024<\/a>.<\/p>\n
\n- **\u6975(Pole)**\uff1a$F(s)$ \u306e\u5206\u6bcd\u3092\u30bc\u30ed\u306b\u3057\u3001\u5909\u63db\u3092\u7121\u9650\u5927\u306b\u767a\u6563\u3055\u305b\u308b\u70b9\uff1b
\n- **\u96f6\u70b9(Zero)**\uff1a$F(s)$ \u306e\u5206\u5b50\u3092\u30bc\u30ed\u306b\u3057\u3001\u5909\u63db\u3092\u30bc\u30ed\u306b\u3059\u308b\u70b9\u3002<\/p>\n
\n```<\/p>\n
\ntitle: \u4f8b\u984c 7.2 \u96e2\u6563\u7cfb\u5217\u306eZ\u5909\u63db \u2014 \u53ce\u675f\u9818\u57df\u3068\u5b89\u5b9a\u6027\u89e3\u6790 (Example 7.2 Z-Transform of a Discrete Sequence \u2014 ROC and Stability Analysis)<\/p>\n
\n```<\/p>\n
\n- \u6975\u304c\u865a\u8ef8\u4e0a\uff08$\\text{Re}(s) = 0$\uff09\u306b\u3042\u308b\uff1a\u30b7\u30b9\u30c6\u30e0\u306f\u81e8\u754c\u5b89\u5b9a\u3001\u30a4\u30f3\u30d1\u30eb\u30b9\u5fdc\u7b54\u306f\u7b49\u632f\u5e45\u632f\u52d5\u3002<\/p>\n
\n$z$ \u5e73\u9762\u4e0a\u3067\u6975\u3068\u96f6\u70b9\u3092\u914d\u7f6e\u3057\u3001\u76ee\u6a19\u306e\u5468\u6ce2\u6570\u5fdc\u7b54\u306b\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n
\ntitle: \u5b9a\u7fa9 8.1 \u7573\u307f\u8fbc\u307f (Definition 8.1 Convolution)
\n$f, g: \\mathbb{R} \\to \\mathbb{R}$ \u3092\u4e8c\u3064\u306e\u9023\u7d9a\u95a2\u6570\u3068\u3059\u308b\u3002\u305d\u306e**\u7573\u307f\u8fbc\u307f(Convolution)** \u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b:<\/p>\n
\n```<\/p>\n
\ntitle: \u547d\u984c 8.1 \u7573\u307f\u8fbc\u307f\u306e\u5185\u7a4d\u89e3\u91c8 (Proposition 8.1 Inner Product Interpretation of Convolution)
\n\u56fa\u5b9a\u6642\u523b $t$ \u306b\u304a\u3044\u3066\u3001\u7573\u307f\u8fbc\u307f\u6f14\u7b97 $(f * g)(t)$ \u306f\u95a2\u6570 $f(\\tau)$ \u3068\u53cd\u8ee2\u30fb\u5e73\u884c\u79fb\u52d5\u3055\u308c\u305f $g(\\tau)$ \u3068\u306e\u9593\u306e\u5185\u7a4d\u306b\u7b49\u3057\u3044:<\/p>\n
\n```<\/p>\n
\ntitle: \u5b9a\u7fa9 8.2 \u76f8\u4e92\u76f8\u95a2 (Definition 8.2 Cross-Correlation)
\n\u7573\u307f\u8fbc\u307f\u3068\u5bc6\u63a5\u306b\u95a2\u9023\u3059\u308b\u6f14\u7b97\u3068\u3057\u3066**\u76f8\u4e92\u76f8\u95a2(Cross-Correlation)** \u304c\u3042\u308b:<\/p>\n
\n```<\/p>\n
\n2. **\u5e73\u884c\u79fb\u52d5(Shift)**\uff1a\u53cd\u8ee2\u3057\u305f\u30ab\u30fc\u30cd\u30eb\u3092 $t$ \u3060\u3051\u5e73\u884c\u79fb\u52d5\u3057\u3001$g(t - \\tau)$ \u3092\u5f97\u308b\uff1b
\n3. **\u4e57\u7b97(Multiply)**\uff1a$f(\\tau)$ \u3068 $g(t - \\tau)$ \u3092\u70b9\u3054\u3068\u306b\u4e57\u7b97\u3059\u308b\uff1b
\n4. **\u7a4d\u5206(Integrate)**\uff1a\u4e57\u7b97\u7d50\u679c\u3092\u7dcf\u548c\uff08\u7a4d\u5206\uff09\u3057\u3001\u305d\u306e\u6642\u523b\u306b\u304a\u3051\u308b\u5185\u7a4d\u5024\u3092\u5f97\u308b\u3002<\/p>\n
\ntitle: \u4f8b\u984c 8.1 \u96e2\u6563\u7cfb\u5217\u306e\u6ed1\u52d5\u5185\u7a4d\u7573\u307f\u8fbc\u307f \u2014 \u4e00\u70b9\u305a\u3064\u306e\u624b\u8a08\u7b97 (Example 8.1 Sliding Inner Product Convolution of Discrete Sequences \u2014 Point-by-Point Manual Calculation)<\/p>\n
\n$$y[0] = \\sum_{k} x[k]h[0-k] = x[0]h[0] = 1 \\times 0.5 = 0.5$$<\/p>\n
\n$$y[1] = x[0]h[1] + x[1]h[0] = 1 \\times 1 + 2 \\times 0.5 = 2$$<\/p>\n
\n$$y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 1 \\times 0.5 + 2 \\times 1 + 3 \\times 0.5 = 4$$<\/p>\n
\n$$y[3] = x[1]h[2] + x[2]h[1] = 2 \\times 0.5 + 3 \\times 1 = 4$$<\/p>\n
\n$$y[4] = x[2]h[2] = 3 \\times 0.5 = 1.5$$<\/p>\n
\n```<\/p>\n
\ntitle: \u4f8b\u984c 8.2 Sobel \u30a8\u30c3\u30b8\u691c\u51fa \u2014 \u5185\u7a4d\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u3068\u3057\u3066\u306e\u4e8c\u6b21\u5143\u7573\u307f\u8fbc\u307f (Example 8.2 Sobel Edge Detection \u2014 2D Convolution as Inner Product Template)<\/p>\n
\n```<\/p>\n![](\"https:\/\/r2.wuhanqing.cn\/MyWebsiteFiles\/1-%E6%96%87%E7%AB%A0\/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0\/%E4%BB%8E%E7%82%B9%E7%A7%AF%E5%88%B0%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4%EF%BC%9A%E8%97%8F%E5%9C%A8%E5%BE%AE%E7%A7%AF%E5%88%86%E3%80%81%E4%BF%A1%E5%8F%B7%E4%B8%8EAI%E8%83%8C%E5%90%8E%E7%9A%84%E5%90%8C%E4%B8%80%E5%A5%97%E8%AF%AD%E8%A8%80\/Pictures\/04_convolution_matched_filter.png\")
<\/p>\n![](\"https:\/\/r2.wuhanqing.cn\/MyWebsiteFiles\/1-%E6%96%87%E7%AB%A0\/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0\/%E4%BB%8E%E7%82%B9%E7%A7%AF%E5%88%B0%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4%EF%BC%9A%E8%97%8F%E5%9C%A8%E5%BE%AE%E7%A7%AF%E5%88%86%E3%80%81%E4%BF%A1%E5%8F%B7%E4%B8%8EAI%E8%83%8C%E5%90%8E%E7%9A%84%E5%90%8C%E4%B8%80%E5%A5%97%E8%AF%AD%E8%A8%80\/Pictures\/05_sobel_edge_detection.png\")
<\/p>\n
\ntitle: \u5b9a\u7fa9 9.1 \u4e8c\u6b21\u5143DCT (Definition 9.1 2D DCT)
\n$f(x, y)$ \u3092 $N \\times N$ \u306e\u753b\u50cf\u30d6\u30ed\u30c3\u30af\uff08$x, y = 0, 1, \\dots, N-1$\uff09\u3068\u3059\u308b\u3002\u305d\u306e\u4e8c\u6b21\u5143DCT\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b:<\/p>\n
\n```<\/p>\n
\ntitle: \u547d\u984c 9.1 \u76f4\u4ea4\u5c04\u5f71\u3068\u3057\u3066\u306eDCT (Proposition 9.1 DCT as Orthogonal Projection)
\n$N \\times N$ \u500b\u306eDCT\u57fa\u5e95\u95a2\u6570\u3092\u5b9a\u7fa9\u3059\u308b:<\/p>\n
\n```<\/p>\n
\ntitle: \u547d\u984c 9.2 \u30a8\u30cd\u30eb\u30ae\u30fc\u96c6\u4e2d\u6027 (Proposition 9.2 Energy Compaction)
\n\u81ea\u7136\u753b\u50cf\u306b\u304a\u3044\u3066\u3001DCT\u4fc2\u6570\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u4e3b\u306b\u4f4e\u5468\u6ce2\u9818\u57df\uff08$u, v$ \u304c\u5c0f\u3055\u3044\uff09\u306b\u96c6\u4e2d\u3057\u3001\u9ad8\u5468\u6ce2\u4fc2\u6570\uff08$u, v$ \u304c\u5927\u304d\u3044\uff09\u306e\u632f\u5e45\u306f\u30bc\u30ed\u306b\u8fd1\u3065\u304f\u3002JPEG\u5727\u7e2e\u306f\u3053\u306e\u7279\u6027\u3092\u5229\u7528\u3057\u3001\u91cf\u5b50\u5316\u306b\u3088\u3063\u3066\u5fae\u5c0f\u306a\u9ad8\u5468\u6ce2\u4fc2\u6570\u3092\u5207\u308a\u6368\u3066\u308b\u3053\u3068\u3067\u3001\u8996\u899a\u54c1\u8cea\u3092\u7dad\u6301\u3057\u3064\u3064\u5927\u5e45\u306a\u5727\u7e2e\u3092\u5b9f\u73fe\u3059\u308b\u3002
\n```<\/p>\n
\n- **\u4f4e\u5468\u6ce2\u57fa\u5e95**\uff08$u, v$ \u304c\u5c0f\u3055\u3044\uff09\uff1a\u6ed1\u3089\u304b\u306a\u52fe\u914d\u30d1\u30bf\u30fc\u30f3\u3001\u753b\u50cf\u306e\u5927\u898f\u6a21\u69cb\u9020\u306b\u5bfe\u5fdc\uff1b
\n- **\u9ad8\u5468\u6ce2\u57fa\u5e95**\uff08$u, v$ \u304c\u5927\u304d\u3044\uff09\uff1a\u5bc6\u306a\u632f\u52d5\u30d1\u30bf\u30fc\u30f3\u3001\u753b\u50cf\u306e\u8a73\u7d30\u306a\u30c6\u30af\u30b9\u30c1\u30e3\u3068\u30ce\u30a4\u30ba\u306b\u5bfe\u5fdc\u3002<\/p>\n
\ntitle: \u4f8b\u984c 9.1 $2 \\times 2$ \u753b\u50cf\u30d6\u30ed\u30c3\u30af\u306eDCT\u5c04\u5f71\u4fc2\u6570\u306e\u624b\u8a08\u7b97 (Example 9.1 Manual Calculation of DCT Projection Coefficients for a $2 \\times 2$ Image Block)<\/p>\n
\n- $F(0,1) = 20$\uff1a\u6c34\u5e73\u65b9\u5411\u306e\u9ad8\u5468\u6ce2\u6210\u5206\u3001\u5de6\u53f3\u306e\u30d4\u30af\u30bb\u30eb\u5dee\u3092\u53cd\u6620\u3002
\n- $F(1,0) = 40$\uff1a\u5782\u76f4\u65b9\u5411\u306e\u9ad8\u5468\u6ce2\u6210\u5206\u3001\u4e0a\u4e0b\u306e\u30d4\u30af\u30bb\u30eb\u5dee\u3092\u53cd\u6620\u3002
\n- $F(1,1) = 0$\uff1a\u5bfe\u89d2\u65b9\u5411\u306e\u9ad8\u5468\u6ce2\u6210\u5206\u3001\u30bc\u30ed\u3067\u3042\u308b\u305f\u3081\u5bfe\u89d2\u30c6\u30af\u30b9\u30c1\u30e3\u304c\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n
\n```<\/p>\n
\n2. **DCT\u5909\u63db**\uff1a\u5404\u30d6\u30ed\u30c3\u30af\u306b\u5bfe\u3057\u3066\u4e8c\u6b21\u5143DCT\u3092\u5b9f\u884c\u3057\u300164\u500b\u306e\u5468\u6ce2\u6570\u9818\u57df\u4fc2\u6570\u3092\u5f97\u308b\uff1b
\n3. **\u91cf\u5b50\u5316**\uff1a\u91cf\u5b50\u5316\u884c\u5217\u3067DCT\u4fc2\u6570\u3092\u9664\u7b97\u3057\uff08\u9ad8\u5468\u6ce2\u307b\u3069\u91cf\u5b50\u5316\u30b9\u30c6\u30c3\u30d7\u304c\u5927\u304d\u3044\uff09\u3001\u5fae\u5c0f\u306a\u4fc2\u6570\u3092\u30bc\u30ed\u306b\u8a2d\u5b9a\u3059\u308b\uff1b
\n4. **\u30a8\u30f3\u30c8\u30ed\u30d4\u30fc\u7b26\u53f7\u5316**\uff1a\u91cf\u5b50\u5316\u5f8c\u306e\u4fc2\u6570\u306b\u5bfe\u3057\u3066\u30cf\u30d5\u30de\u30f3\u7b26\u53f7\u5316\u307e\u305f\u306f\u7b97\u8853\u7b26\u53f7\u5316\u3092\u65bd\u3059\u3002<\/p>\n
\ntitle: \u5b9a\u7fa9 10.1 \u77ed\u6642\u9593\u30d5\u30fc\u30ea\u30a8\u5909\u63db (Definition 10.1 Short-Time Fourier Transform)
\n\u6642\u9593\u5b9a\u4f4d\u306e\u6b20\u5982\u3092\u88dc\u3046\u305f\u3081\u3001\u77ed\u6642\u9593\u30d5\u30fc\u30ea\u30a8\u5909\u63db(STFT: Short-Time Fourier Transform)\u306f\u7a93\u95a2\u6570 $w(t)$ \u3092\u5c0e\u5165\u3059\u308b:<\/p>\n
\n```<\/p>\n
\ntitle: \u5b9a\u7fa9 10.2 \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db (Definition 10.2 Wavelet Transform)
\n\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u306f\u3001\u4f38\u7e2e\u30fb\u5e73\u884c\u79fb\u52d5\u53ef\u80fd\u306a\u57fa\u5e95\u95a2\u6570 $\\psi_{a,b}(t)$ \u3092\u7528\u3044\u308b\u3053\u3068\u3067\u3001\u6642\u5468\u6ce2\u6570\u5206\u89e3\u80fd\u306e\u77db\u76fe\u3092\u6839\u672c\u7684\u306b\u89e3\u6c7a\u3059\u308b$^{[17]}$\u3002$\\psi(t)$ \u3092\u30de\u30b6\u30fc\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8(Mother Wavelet)\u3068\u3057\u3001$\\int \\psi(t)\\,dt = 0$\uff08\u30bc\u30ed\u5e73\u5747\u6761\u4ef6\uff09\u3092\u6e80\u305f\u3059\u3082\u306e\u3068\u3059\u308b\u3002\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u57fa\u5e95\u95a2\u6570\u65cf\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b:<\/p>\n
\n```<\/p>\n
\ntitle: \u5b9a\u7fa9 10.3 \u9023\u7d9a\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db (Definition 10.3 Continuous Wavelet Transform)
\n\u4fe1\u53f7 $f(t)$ \u306e\u9023\u7d9a\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db(CWT: Continuous Wavelet Transform)\u306f\u3001$f$ \u3068\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u57fa\u5e95\u95a2\u6570\u306e\u5185\u7a4d\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b:<\/p>\n
\n```<\/p>\n
\ntitle: \u547d\u984c 10.1 \u30de\u30eb\u30c1\u30ec\u30be\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u89e3\u6790 (Proposition 10.1 Multi-Resolution Analysis)
\n\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u306e\u6642\u5468\u6ce2\u6570\u5206\u89e3\u80fd\u306f\u30b9\u30b1\u30fc\u30eb $a$ \u306b\u5fdc\u3058\u3066\u9069\u5fdc\u7684\u306b\u5909\u5316\u3059\u308b:<\/p>\n
\n- **\u5927\u30b9\u30b1\u30fc\u30eb $a$**\uff08\u4f4e\u5468\u6ce2\uff09\uff1a\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u306f\u4f38\u9577\u3055\u308c\u3001\u5468\u6ce2\u6570\u5206\u89e3\u80fd\u304c\u9ad8\u304f\u6642\u9593\u5206\u89e3\u80fd\u304c\u4f4e\u3044\u3002\u9577\u671f\u7684\u306a\u30c8\u30ec\u30f3\u30c9\u306e\u89e3\u6790\u306b\u9069\u3059\u308b\u3002<\/p>\n
\n```<\/p>\n
\n- **\u5c0f\u3055\u306a\u30d7\u30ed\u30fc\u30d6\uff08\u5c0f\u30b9\u30b1\u30fc\u30eb $a$\uff09**\uff1a\u72ed\u3044\u6642\u9593\u7bc4\u56f2\u3092\u30ab\u30d0\u30fc\u3057\u3001\u4fe1\u53f7\u306e\u6025\u5909\u70b9\uff08\u9ad8\u5468\u6ce2\uff09\u3092\u6b63\u78ba\u306b\u7279\u5b9a\u3059\u308b\u304c\u3001\u5168\u4f53\u7684\u306a\u30c8\u30ec\u30f3\u30c9\u306f\u898b\u3048\u306a\u3044\u3002<\/p>\n
\ntitle: \u4f8b\u984c 10.1 Haar \u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5206\u89e3 \u2014 \u4e00\u6bb5\u76ee\u3068\u4e8c\u6bb5\u76ee\u306e\u30a6\u30a7\u30fc\u30d6\u30ec\u30c3\u30c8\u5909\u63db\u306e\u624b\u8a08\u7b97 (Example 10.1 Haar Wavelet Decomposition \u2014 Manual Calculation of Level-1 and Level-2 Wavelet Transforms)<\/p>\n
\n2. **\u6700\u5c0f\u4e8c\u4e57\u6cd5**\uff1a`np.linalg.lstsq` \u3092\u5229\u7528\u3057\u3066\u6b63\u898f\u65b9\u7a0b\u5f0f $A^T A \\hat{x} = A^T b$\uff08\u5b9a\u7406 3.1\uff09\u3092\u89e3\u304f\u3002\u672c\u8cea\u7684\u306b\u306f\u89b3\u6e2c\u30d9\u30af\u30c8\u30eb\u3092\u30e2\u30c7\u30eb\u7a7a\u9593\u306b\u76f4\u4ea4\u5c04\u5f71\u3057\u3066\u3044\u308b\u3002
\n3. **\u30d5\u30fc\u30ea\u30a8\u5206\u89e3**\uff1aFFT\u306b\u3088\u308a\u6642\u9593\u9818\u57df\u4fe1\u53f7\u3092\u5468\u6ce2\u6570\u57fa\u5e95\u306b\u5c04\u5f71\u3059\u308b\uff08\u5b9a\u7406 6.1\uff09\u3002\u30b9\u30da\u30af\u30c8\u30eb\u5185\u306e\u5404\u30d4\u30fc\u30af\u306f\u4e00\u3064\u306e\u5468\u6ce2\u6570\u6210\u5206\u306e\u5185\u7a4d\u4fc2\u6570\u306b\u5bfe\u5fdc\u3059\u308b\u3002
\n4. **\u7573\u307f\u8fbc\u307f\u3068\u30de\u30c3\u30c1\u30c9\u30d5\u30a3\u30eb\u30bf**\uff1a\u7573\u307f\u8fbc\u307f\u3092\u6ed1\u52d5\u5185\u7a4d\u6f14\u7b97\u3068\u898b\u306a\u3057\uff08\u5b9a\u7fa9 8.1\uff09\u3001\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u3068\u4fe1\u53f7\u3092\u70b9\u3054\u3068\u306b\u5185\u7a4d\u3057\u3066\u30d1\u30eb\u30b9\u4f4d\u7f6e\u3092\u691c\u51fa\u3059\u308b\u3002
\n5. **Sobel\u30a8\u30c3\u30b8\u691c\u51fa**\uff1a\u4e8c\u6b21\u5143\u7573\u307f\u8fbc\u307f\u30ab\u30fc\u30cd\u30eb\u3068\u753b\u50cf\u306e\u5185\u7a4d\u3092\u8a08\u7b97\u3057\uff08\u4f8b\u984c 8.2\uff09\u3001\u5404\u30d4\u30af\u30bb\u30eb\u306b\u304a\u3051\u308b\u52fe\u914d\u632f\u5e45\u3092\u6c42\u3081\u308b\u3002<\/p>\ndef cosine_similarity(vec_a: np.ndarray, vec_b: np.ndarray) -> float:\r\n dot_product = float(np.dot(vec_a, vec_b))\r\n norm_a = np.linalg.norm(vec_a)\r\n norm_b = np.linalg.norm(vec_b)\r\n return dot_product \/ (norm_a * norm_b)\r\n\r\ndef build_semantic_demo() -> tuple[list[str], dict[str, np.ndarray], np.ndarray]:\r\n tokens = [\"king\", \"queen\", \"man\", \"woman\", \"apple\"]\r\n embeddings = {\r\n \"king\": np.array([0.92, 0.10, 0.78, 0.25, 0.60]),\r\n \"queen\": np.array([0.90, 0.12, 0.80, 0.30, 0.63]),\r\n \"man\": np.array([0.88, 0.18, 0.40, 0.22, 0.35]),\r\n \"woman\": np.array([0.86, 0.22, 0.42, 0.28, 0.38]),\r\n \"apple\": np.array([0.05, 0.95, 0.08, 0.87, 0.10]),\r\n }\r\n matrix = np.array(\r\n [[cosine_similarity(embeddings[left], embeddings[right]) for right in tokens] for left in tokens]\r\n )\r\n return tokens, embeddings, matrix\r\n<\/code><\/pre>\n