94 lines
3.4 KiB
Plaintext
94 lines
3.4 KiB
Plaintext
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"source": [
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"# 符号\n",
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":label:`chap_notation`\n",
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"\n",
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"本书中使用的符号概述如下。\n",
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"\n",
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"## 数字\n",
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"\n",
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"* $x$：标量\n",
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"* $\\mathbf{x}$：向量\n",
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"* $\\mathbf{X}$：矩阵\n",
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"* $\\mathsf{X}$：张量\n",
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"* $\\mathbf{I}$：单位矩阵\n",
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"* $x_i$, $[\\mathbf{x}]_i$：向量$\\mathbf{x}$第$i$个元素\n",
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"* $x_{ij}$, $[\\mathbf{X}]_{ij}$：矩阵$\\mathbf{X}$第$i$行第$j$列的元素\n",
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"\n",
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"## 集合论\n",
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"\n",
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"* $\\mathcal{X}$: 集合\n",
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"* $\\mathbb{Z}$: 整数集合\n",
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"* $\\mathbb{R}$: 实数集合\n",
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"* $\\mathbb{R}^n$: $n$维实数向量集合\n",
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"* $\\mathbb{R}^{a\\times b}$: 包含$a$行和$b$列的实数矩阵集合\n",
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"* $\\mathcal{A}\\cup\\mathcal{B}$: 集合$\\mathcal{A}$和$\\mathcal{B}$的并集\n",
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"* $\\mathcal{A}\\cap\\mathcal{B}$：集合$\\mathcal{A}$和$\\mathcal{B}$的交集\n",
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"* $\\mathcal{A}\\setminus\\mathcal{B}$：集合$\\mathcal{A}$与集合$\\mathcal{B}$相减，$\\mathcal{B}$关于$\\mathcal{A}$的相对补集\n",
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"\n",
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"## 函数和运算符\n",
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"\n",
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"* $f(\\cdot)$：函数\n",
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"* $\\log(\\cdot)$：自然对数\n",
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"* $\\exp(\\cdot)$: 指数函数\n",
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"* $\\mathbf{1}_\\mathcal{X}$: 指示函数\n",
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"* $\\mathbf{(\\cdot)}^\\top$: 向量或矩阵的转置\n",
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"* $\\mathbf{X}^{-1}$: 矩阵的逆\n",
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"* $\\odot$: 按元素相乘\n",
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"* $[\\cdot, \\cdot]$：连结\n",
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"* $\\lvert \\mathcal{X} \\rvert$：集合的基数\n",
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"* $\\|\\cdot\\|_p$: ：$L_p$ 正则\n",
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"* $\\|\\cdot\\|$: $L_2$ 正则\n",
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"* $\\langle \\mathbf{x}, \\mathbf{y} \\rangle$：向量$\\mathbf{x}$和$\\mathbf{y}$的点积\n",
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"* $\\sum$: 连加\n",
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"* $\\prod$: 连乘\n",
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"* $\\stackrel{\\mathrm{def}}{=}$：定义\n",
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"\n",
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"## 微积分\n",
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"\n",
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"* $\\frac{dy}{dx}$：$y$关于$x$的导数\n",
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"* $\\frac{\\partial y}{\\partial x}$：$y$关于$x$的偏导数\n",
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"* $\\nabla_{\\mathbf{x}} y$：$y$关于$\\mathbf{x}$的梯度\n",
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"* $\\int_a^b f(x) \\;dx$: $f$在$a$到$b$区间上关于$x$的定积分\n",
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"* $\\int f(x) \\;dx$: $f$关于$x$的不定积分\n",
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"\n",
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"## 概率与信息论\n",
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"\n",
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"* $P(\\cdot)$：概率分布\n",
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"* $z \\sim P$: 随机变量$z$具有概率分布$P$\n",
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"* $P(X \\mid Y)$：$X\\mid Y$的条件概率\n",
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"* $p(x)$: 概率密度函数\n",
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"* ${E}_{x} [f(x)]$: 函数$f$对$x$的数学期望\n",
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"* $X \\perp Y$: 随机变量$X$和$Y$是独立的\n",
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"* $X \\perp Y \\mid Z$: 随机变量$X$和$Y$在给定随机变量$Z$的条件下是独立的\n",
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"* $\\mathrm{Var}(X)$: 随机变量$X$的方差\n",
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"* $\\sigma_X$: 随机变量$X$的标准差\n",
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"* $\\mathrm{Cov}(X, Y)$: 随机变量$X$和$Y$的协方差\n",
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"* $\\rho(X, Y)$: 随机变量$X$和$Y$的相关性\n",
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"* $H(X)$: 随机变量$X$的熵\n",
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"* $D_{\\mathrm{KL}}(P\\|Q)$: $P$和$Q$的KL-散度\n",
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"\n",
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"## 复杂度\n",
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"\n",
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"* $\\mathcal{O}$：大O标记\n",
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"\n",
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"[Discussions](https://discuss.d2l.ai/t/2089)\n"
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]
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