Notation
:label:chap_notation
The notation used throughout this book is summarized below.
Numbers
- $x$: A scalar
- $\mathbf{x}$: A vector
- $\mathbf{X}$: A matrix
- $\mathsf{X}$: A tensor
- $\mathbf{I}$: An identity matrix
- $x_i$, $[\mathbf{x}]_i$: The $i^\mathrm{th}$ element of vector $\mathbf{x}$
- $x{ij}$, $[\mathbf{X}]{ij}$: The element of matrix $\mathbf{X}$ at row $i$ and column $j$
Set Theory
- $\mathcal{X}$: A set
- $\mathbb{Z}$: The set of integers
- $\mathbb{R}$: The set of real numbers
- $\mathbb{R}^n$: The set of $n$-dimensional vectors of real numbers
- $\mathbb{R}^{a\times b}$: The set of matrices of real numbers 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 $\mathcal{B}$
- $\mathcal{A}\setminus\mathcal{B}$: Subtraction of set $\mathcal{B}$ from set $\mathcal{A}$
Functions and Operators
- $f(\cdot)$: A function
- $\log(\cdot)$: The natural logarithm
- $\exp(\cdot)$: The exponential function
- $\mathbf{1}_\mathcal{X}$: The indicator function
- $\mathbf{(\cdot)}^\top$: Transpose of a vector or a matrix
- $\mathbf{X}^{-1}$: Inverse of matrix $\mathbf{X}$
- $\odot$: Hadamard (elementwise) product
- $[\cdot, \cdot]$: Concatenation
- $\lvert \mathcal{X} \rvert$: Cardinality of set $\mathcal{X}$
- $|\cdot|_p$: $L_p$ norm
- $|\cdot|$: $L_2$ norm
- $\langle \mathbf{x}, \mathbf{y} \rangle$: Dot product of vectors $\mathbf{x}$ and $\mathbf{y}$
- $\sum$: Series addition
- $\prod$: Series multiplication
- $\stackrel{\mathrm{def}}{=}$: Definition
Calculus
- $\frac{dy}{dx}$: 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 $x$
Probability and Information Theory
- $P(\cdot)$: Probability distribution
- $z \sim P$: Random variable $z$ has probability distribution $P$
- $P(X \mid Y)$: Conditional probability of $X \mid Y$
- $p(x)$: Probability density function
- ${E}_{x} [f(x)]$: Expectation of $f$ 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 random variable $Z$
- $\mathrm{Var}(X)$: 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 $X$
- $D_{\mathrm{KL}}(P|Q)$: KL-divergence of distributions $P$ and $Q$
Complexity
- $\mathcal{O}$: Big O notation