数字符号概述

Numbers

\(x\): Scalar
\(\mathbf{x}\): Vector
\(\mathbf{X}\): Matrix
\(\mathsf{X}\): Tensor
\(\mathbf{I}\): Identity matrix
\(x_i\), \([\mathbf{x}]_i\): \(i\)-th element of vector \(\mathbf{x}\)
\(x_{ij}\), \([\mathbf{X}]_{ij}\): Element at \(i\)-th row, \(j\)-th column of matrix \(\mathbf{X}\)

\(\mathcal{X}\): Set
\(\mathbb{Z}\): Set of integers
\(\mathbb{R}\): Set of real numbers
\(\mathbb{R}^n\): Set of \(n\)-dimensional real vectors
\(\mathbb{R}^{a\times b}\): Set of real matrices with \(a\) rows and \(b\) columns
\(\mathcal{A}\cup\mathcal{B}\): Union of sets \(\mathcal{A}\) and \(\mathcal{B}\)
\(\mathcal{A}\cap\mathcal{B}\): Intersection of sets \(\mathcal{A}\) and \(\mathcal{B}\)
\(\mathcal{A}\setminus\mathcal{B}\): Relative complement of set \(\mathcal{B}\) in set \(\mathcal{A}\)

\(f(\cdot)\): Function
\(\log(\cdot)\): Natural logarithm
\(\exp(\cdot)\): Exponential function
\(\mathbf{1}_\mathcal{X}\): Indicator function
\((\cdot)^\top\): Transpose of vector or matrix
\(\mathbf{X}^{-1}\): Inverse of matrix
\(\odot\): Element-wise multiplication
\([\cdot, \cdot]\): Concatenation
\(|\mathcal{X}|\): Cardinality of set
\(\|\cdot\|_p\): \(L_p\) norm
\(\|\cdot\|\): \(L_2\) norm
\(\langle \mathbf{x}, \mathbf{y} \rangle\): Dot product of vectors \(\mathbf{x}\) and \(\mathbf{y}\)
\(\sum\): Summation
\(\prod\): Product
\(\stackrel{\mathrm{def}}{=}\): Definition

\(\frac{dy}{dx}\): Derivative of \(y\) with respect to \(x\)
\(\frac{\partial y}{\partial x}\): Partial derivative of \(y\) with respect to \(x\)
\(\nabla_{\mathbf{x}} y\): Gradient of \(y\) with respect to \(\mathbf{x}\)
\(\int_a^b f(x) \;dx\): Definite integral of \(f\) from \(a\) to \(b\) with respect to \(x\)
\(\int f(x) \;dx\): Indefinite integral of \(f\) with respect to \(x\)

\(P(\cdot)\): Probability distribution
\(z \sim P\): Random variable \(z\) follows distribution \(P\)
\(P(X \mid Y)\): Conditional probability of \(X\) given \(Y\)
\(p(x)\): Probability density function
\({E}_{x} [f(x)]\): Expectation of function \(f\) with respect to \(x\)
\(X \perp Y\): Random variables \(X\) and \(Y\) are independent
\(X \perp Y \mid Z\): Random variables \(X\) and \(Y\) are conditionally independent given \(Z\)
\(\mathrm{Var}(X)\): Variance of random variable \(X\)
\(\sigma_X\): Standard deviation of random variable \(X\)
\(\mathrm{Cov}(X, Y)\): Covariance of random variables \(X\) and \(Y\)
\(\rho(X, Y)\): Correlation of random variables \(X\) and \(Y\)
\(H(X)\): Entropy of random variable \(X\)
\(D_{\mathrm{KL}}(P\|Q)\): Kullback-Leibler divergence between \(P\) and \(Q\)