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5장 수식 번역

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Haesun Park
2018-04-25 22:43:39 +09:00
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book_equations.ipynb
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@@ -355,78 +355,78 @@
},
"cell_type": "markdown",
"source": [
"# Chapter 5\n",
"**Equation 5-1: Gaussian RBF**\n",
"# 5장\n",
"**식 5-1: 가우시안 RBF**\n",
"\n",
"$\n",
"{\\displaystyle \\phi_{\\gamma}(\\mathbf{x}, \\mathbf{\\ell})} = {\\displaystyle \\exp({\\displaystyle -\\gamma \\left\\| \\mathbf{x} - \\mathbf{\\ell} \\right\\|^2})}\n",
"$\n",
"\n",
"\n",
"**Equation 5-2: Linear SVM classifier prediction**\n",
"**식 5-2: 선형 SVM 분류기의 예측**\n",
"\n",
"$\n",
"\\hat{y} = \\begin{cases}\n",
" 0 & \\text{if } \\mathbf{w}^T \\cdot \\mathbf{x} + b < 0, \\\\\n",
" 1 & \\text{if } \\mathbf{w}^T \\cdot \\mathbf{x} + b \\geq 0\n",
" 0 & \\mathbf{w}^T \\cdot \\mathbf{x} + b < 0 \\text{일 때 } \\\\\n",
" 1 & \\mathbf{w}^T \\cdot \\mathbf{x} + b \\geq 0 \\text{일 때 }\n",
"\\end{cases}\n",
"$\n",
"\n",
"\n",
"**Equation 5-3: Hard margin linear SVM classifier objective**\n",
"**식 5-3: 하드 마진 선형 SVM 분류기의 목적 함수**\n",
"\n",
"$\n",
"\\begin{split}\n",
"&\\underset{\\mathbf{w}, b}{\\operatorname{minimize}}\\quad{\\frac{1}{2}\\mathbf{w}^T \\cdot \\mathbf{w}} \\\\\n",
"&\\text{subject to} \\quad t^{(i)}(\\mathbf{w}^T \\cdot \\mathbf{x}^{(i)} + b) \\ge 1 \\quad \\text{for } i = 1, 2, \\dots, m\n",
"&\\underset{\\mathbf{w}, b}{\\operatorname{minimize}}\\,{\\frac{1}{2}\\mathbf{w}^T \\cdot \\mathbf{w}} \\\\\n",
"&[\\text{조건}] \\, i = 1, 2, \\dots, m \\text{일 때} \\quad t^{(i)}(\\mathbf{w}^T \\cdot \\mathbf{x}^{(i)} + b) \\ge 1\n",
"\\end{split}\n",
"$\n",
"\n",
"\n",
"**Equation 5-4: Soft margin linear SVM classifier objective**\n",
"**식 5-4: 소프트 마진 선형 SVM 분류기의 목적 함수**\n",
"\n",
"$\n",
"\\begin{split}\n",
"&\\underset{\\mathbf{w}, b, \\mathbf{\\zeta}}{\\operatorname{minimize}}\\quad{\\dfrac{1}{2}\\mathbf{w}^T \\cdot \\mathbf{w} + C \\sum\\limits_{i=1}^m{\\zeta^{(i)}}}\\\\\n",
"&\\text{subject to} \\quad t^{(i)}(\\mathbf{w}^T \\cdot \\mathbf{x}^{(i)} + b) \\ge 1 - \\zeta^{(i)} \\quad \\text{and} \\quad \\zeta^{(i)} \\ge 0 \\quad \\text{for } i = 1, 2, \\dots, m\n",
"&\\underset{\\mathbf{w}, b, \\mathbf{\\zeta}}{\\operatorname{minimize}}\\,{\\dfrac{1}{2}\\mathbf{w}^T \\cdot \\mathbf{w} + C \\sum\\limits_{i=1}^m{\\zeta^{(i)}}}\\\\\n",
"&[\\text{조건}] \\, i = 1, 2, \\dots, m \\text{일 때} \\quad t^{(i)}(\\mathbf{w}^T \\cdot \\mathbf{x}^{(i)} + b) \\ge 1 - \\zeta^{(i)} \\text{ 이고} \\quad \\zeta^{(i)} \\ge 0\n",
"\\end{split}\n",
"$\n",
"\n",
"\n",
"**Equation 5-5: Quadratic Programming problem**\n",
"**식 5-5: QP 문제**\n",
"\n",
"$\n",
"\\begin{split}\n",
"\\underset{\\mathbf{p}}{\\text{Minimize}} \\quad & \\dfrac{1}{2} \\mathbf{p}^T \\cdot \\mathbf{H} \\cdot \\mathbf{p} \\quad + \\quad \\mathbf{f}^T \\cdot \\mathbf{p} \\\\\n",
"\\text{subject to} \\quad & \\mathbf{A} \\cdot \\mathbf{p} \\le \\mathbf{b} \\\\\n",
"\\text{where } &\n",
"\\underset{\\mathbf{p}}{\\text{minimize}} \\, & \\dfrac{1}{2} \\mathbf{p}^T \\cdot \\mathbf{H} \\cdot \\mathbf{p} \\, + \\, \\mathbf{f}^T \\cdot \\mathbf{p} \\\\\n",
"[\\text{조건}] \\, & \\mathbf{A} \\cdot \\mathbf{p} \\le \\mathbf{b} \\\\\n",
"\\text{여기서 } &\n",
"\\begin{cases}\n",
" \\mathbf{p} & \\text{ is an }n_p\\text{-dimensional vector (} n_p = \\text{number of parameters),}\\\\\n",
" \\mathbf{H} & \\text{ is an }n_p \\times n_p \\text{ matrix,}\\\\\n",
" \\mathbf{f} & \\text{ is an }n_p\\text{-dimensional vector,}\\\\\n",
" \\mathbf{A} & \\text{ is an } n_c \\times n_p \\text{ matrix (}n_c = \\text{number of constraints),}\\\\\n",
" \\mathbf{b} & \\text{ is an }n_c\\text{-dimensional vector.}\n",
" \\mathbf{p} \\, \\text{는 }n_p\\text{ 차원의 벡터 (} n_p = \\text{모델 파라미터 수)}\\\\\n",
" \\mathbf{H} \\, \\text{는 }n_p \\times n_p \\text{ 크기 행렬}\\\\\n",
" \\mathbf{f} \\, \\text{는 }n_p\\text{ 차원의 벡터}\\\\\n",
" \\mathbf{A} \\, \\text{는 } n_c \\times n_p \\text{ 크기 행렬 (}n_c = \\text{제약 수)}\\\\\n",
" \\mathbf{b} \\, \\text{는 }n_c\\text{ 차원의 벡터}\n",
"\\end{cases}\n",
"\\end{split}\n",
"$\n",
"\n",
"\n",
"**Equation 5-6: Dual form of the linear SVM objective**\n",
"**식 5-6: 선형 SVM 목적 함수의 쌍대 형식**\n",
"\n",
"$\n",
"\\begin{split}\n",
"\\underset{\\mathbf{\\alpha}}{\\operatorname{minimize}}\n",
"&\\underset{\\mathbf{\\alpha}}{\\operatorname{minimize}} \\,\n",
"\\dfrac{1}{2}\\sum\\limits_{i=1}^{m}{\n",
" \\sum\\limits_{j=1}^{m}{\n",
" \\alpha^{(i)} \\alpha^{(j)} t^{(i)} t^{(j)} {\\mathbf{x}^{(i)}}^T \\cdot \\mathbf{x}^{(j)}\n",
" }\n",
"} \\quad - \\quad \\sum\\limits_{i=1}^{m}{\\alpha^{(i)}}\\\\\n",
"\\text{subject to}\\quad \\alpha^{(i)} \\ge 0 \\quad \\text{for }i = 1, 2, \\dots, m\n",
"} \\, - \\, \\sum\\limits_{i=1}^{m}{\\alpha^{(i)}}\\\\\n",
"&\\text{[조건]}\\,i = 1, 2, \\dots, m \\text{일 때 } \\quad \\alpha^{(i)} \\ge 0\n",
"\\end{split}\n",
"$\n",
"\n",
"\n",
"**Equation 5-7: From the dual solution to the primal solution**\n",
"**식 5-7: 쌍대 문제에서 구한 해로 원 문제의 해 계산하기**\n",
"\n",
"$\n",
"\\begin{split}\n",
@@ -436,7 +436,7 @@
"$\n",
"\n",
"\n",
"**Equation 5-8: Second-degree polynomial mapping**\n",
"**식 5-8: 2차 다항식 매핑**\n",
"\n",
"$\n",
"\\phi\\left(\\mathbf{x}\\right) = \\phi\\left( \\begin{pmatrix}\n",
@@ -450,7 +450,7 @@
"$\n",
"\n",
"\n",
"**Equation 5-9: Kernel trick for a 2^nd^-degree polynomial mapping**\n",
"**식 5-9: 2차 다항식 매핑을 위한 커널 트릭**\n",
"\n",
"$\n",
"\\begin{split}\n",
@@ -473,22 +473,22 @@
"\\end{split}\n",
"$\n",
"\n",
"**In the text about the kernel trick (page 162):**\n",
"[...], then you can replace this dot product of transformed vectors simply by $ ({\\mathbf{x}^{(i)}}^T \\cdot \\mathbf{x}^{(j)})^2 $\n",
"**커널 트릭에 관한 본문 중에서 (220 페이지):**\n",
"[...] 변환된 벡터의 점곱을 간단하게 $ ({\\mathbf{x}^{(i)}}^T \\cdot \\mathbf{x}^{(j)})^2 $ 으로 바꿀 수 있습니다.\n",
"\n",
"\n",
"**Equation 5-10: Common kernels**\n",
"**식 5-10: 일반적인 커널**\n",
"\n",
"$\n",
"\\begin{split}\n",
"\\text{Linear:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\mathbf{a}^T \\cdot \\mathbf{b} \\\\\n",
"\\text{Polynomial:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\left(\\gamma \\mathbf{a}^T \\cdot \\mathbf{b} + r \\right)^d \\\\\n",
"\\text{Gaussian RBF:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\exp({\\displaystyle -\\gamma \\left\\| \\mathbf{a} - \\mathbf{b} \\right\\|^2}) \\\\\n",
"\\text{Sigmoid:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\tanh\\left(\\gamma \\mathbf{a}^T \\cdot \\mathbf{b} + r\\right)\n",
"\\text{선형:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\mathbf{a}^T \\cdot \\mathbf{b} \\\\\n",
"\\text{다항식:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\left(\\gamma \\mathbf{a}^T \\cdot \\mathbf{b} + r \\right)^d \\\\\n",
"\\text{가우시안 RBF:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\exp({\\displaystyle -\\gamma \\left\\| \\mathbf{a} - \\mathbf{b} \\right\\|^2}) \\\\\n",
"\\text{시그모이드:} & \\quad K(\\mathbf{a}, \\mathbf{b}) = \\tanh\\left(\\gamma \\mathbf{a}^T \\cdot \\mathbf{b} + r\\right)\n",
"\\end{split}\n",
"$\n",
"\n",
"**Equation 5-11: Making predictions with a kernelized SVM**\n",
"**식 5-11: 커널 SVM으로 예측하기**\n",
"\n",
"$\n",
"\\begin{split}\n",
@@ -499,7 +499,7 @@
"$\n",
"\n",
"\n",
"**Equation 5-12: Computing the bias term using the kernel trick**\n",
"**식 5-12: 커널 트릭을 사용한 편향 계산**\n",
"\n",
"$\n",
"\\begin{split}\n",
@@ -515,10 +515,10 @@
"$\n",
"\n",
"\n",
"**Equation 5-13: Linear SVM classifier cost function**\n",
"**식 5-13: 선형 SVM 분류기 비용 함수**\n",
"\n",
"$\n",
"J(\\mathbf{w}, b) = \\dfrac{1}{2} \\mathbf{w}^T \\cdot \\mathbf{w} \\quad + \\quad C {\\displaystyle \\sum\\limits_{i=1}^{m}max\\left(0, 1 - t^{(i)}(\\mathbf{w}^T \\cdot \\mathbf{x}^{(i)} + b) \\right)}\n",
"J(\\mathbf{w}, b) = \\dfrac{1}{2} \\mathbf{w}^T \\cdot \\mathbf{w} \\, + \\, C {\\displaystyle \\sum\\limits_{i=1}^{m}max\\left(0, 1 - t^{(i)}(\\mathbf{w}^T \\cdot \\mathbf{x}^{(i)} + b) \\right)}\n",
"$\n",
"\n",
"\n"