# Math macro command for latex support in markdown ### Number and Arrays | command | visualization | comment | | | ------------------ | ------------------ | ---------------------------------------------------------------------------------------------------- | --- | | `a` | $a$ | A scalar | | | `\va` | $\va$ | A vector, additionally $\vzero, \vone, \vmu, \vnu, \vtheta$ for `\vzero, \vone, \vmu, \vnu, \vtheta` | | | `\mA` | $\mA$ | A matrix | | | `\tA` | $\tA$ | A tensor | | | `\mI_n` | $\mI_n$ | Identity matrix with $n$ rows and $n$ columns | | | `\mI` | $\mI$ | Identity matrix with dimensionality implied by context | | | `\ve^{(i)}` | $\ve^{(i)}$ | Standard basis vector $[0,\dots,0,1,0,\dots,0]$ with a 1 at position $i$ | | | `\text{diag}(\va)` | $\text{diag}(\va)$ | A square, diagonal matrix with diagonal entries given by $\va$ | | | `\ra` | $\ra$ | A scalar-valued random variable | | | `\rva` | $\rva$ | A vector-valued random variables | | | `\rmA` | $\rmA$ | A matrix-valued random varialbes | | ### Sets and Graphs | Command | Visualization | Comment | | -------------------- | -------------------- | ------------------------------------------------------------------------------------------- | | `\sA` | $\sA$ | A set
**Note**: the command covers `sA` to `sZ` but don't no `sE` since it's expectation | | `\R` | $\R$ | The set of real numbers | | `{0, 1}` | $\{0, 1\}$ | The set containing 0 and 1 | | `{0, 1, \dots, n}` | $\{0, 1, \dots, n\}$ | The set of all integers between $0$ and $n$ | | `[a, b]` | $[a, b]$ | The real interval including $a$ and $b$ | | `(a, b]` | $(a, b]$ | The real interval excluding $a$ but including $b$ | | `\sA \backslash \sB` | $\sA \backslash \sB$ | Set subtraction, i.e., the set containing the elements of $\sA$ not in $\sB$ | | `\gG` | $\gG$ | A graph | ### Indexing | Command | Visualization | Comment | |------------------------|-------------------------------|-----------------------------------------------------------------| | `\eva_i` | $\eva_i$ | Element $i$ of vector $\va$, with indexing starting at 1 | | `\eva_{-i}` | $\eva_{-i}$ | All elements of vector $\va$ except for element $i$ | | `\emA_{i,j}` | $\emA_{i,j}$ | Element $i, j$ of matrix $\mA$ | | `\mA_{i, :}` | $\mA_{i, :}$ | Row $i$ of matrix $\mA$ | | `\mA_{:, i}` | $\mA_{:, i}$ | Column $i$ of matrix $\mA$ | | `\etA_{i, j, k}` | $\etA_{i, j, k}$ | Element $(i, j, k)$ of a 3-D tensor $\tA$ | | `\tA_{:, :, i}` | $\tA_{:, :, i}$ | 2-D slice of a 3-D tensor | | `\erva_i` | $\erva_i$ | Element $i$ of the random vector $\rva$ | ### Linear Algebra Operators | Command | Visualization | Comment | | ------------------- | ------------------- | -------------------------------------------------- | | `\mA^\top` | $\mA^\top$ | Transpose of matrix $\mA$ | | `\mA^+` | $\mA^+$ | Moore-Penrose pseudoinverse of $\mA$ | | `\mA \odot \mB` | $\mA \odot \mB$ | Element-wise (Hadamard) product of $\mA$ and $\mB$ | | `\mathrm{det}(\mA)` | $\mathrm{det}(\mA)$ | Determinant of $\mA$ | | `\sign(x)` | $\sign(x)$ | Sign of a variable $x$ | | `\Tr \mA` | $\Tr(\mA)$ | Trace of a matrix A | ### Calculus | Command | Visualization | Comment | | ------------------------------------------- | ------------------------------------------- | ---------------------------------------------------------------------- | | `\diff y / \diff x` | $\diff y / \diff x$ | Derivative of $y$ with respect to $x$ | | `\frac{\partial y}{\partial x}` | $\frac{\partial y}{\partial x}$ | Partial derivative of $y$ with respect to $x$ | | `\nabla_\vx y` | $\nabla_\vx y$ | Gradient of $y$ with respect to $\vx$ | | `\nabla_\mX y` | $\nabla_\mX y$ | Matrix derivatives of $y$ with respect to $\mX$ | | `\nabla_\tX y` | $\nabla_\tX y$ | Tensor containing derivatives of $y$ with respect to $\tX$ | | `\frac{\partial f}{\partial \vx}` | $\frac{\partial f}{\partial \vx}$ | Jacobian matrix $\mJ \in \R^{m\times n}$ of $f: \R^n \rightarrow \R^m$ | | `\nabla_\vx^2 f(\vx)\text{ or }\mH(f)(\vx)` | $\nabla_\vx^2 f(\vx)\text{ or }\mH(f)(\vx)$ | The Hessian matrix of $f$ at input point $\vx$ | | `\int f(\vx) d\vx` | $\int f(\vx) d\vx$ | Definite integral over the entire domain of $\vx$ | | `\int_\sS f(\vx) d\vx` | $\int_\sS f(\vx) d\vx$ | Definite integral with respect to $\vx$ over the set $\sS$ | ### Probabilities | Command | Visualization | Comment | |------------------------------------------|-------------------------------------------|-----------------------------------------------------------------------------------------| | `\ra \bot \rb` | $\ra \bot \rb$ | The random variables $\ra$ and $\rb$ are independent | | `\ra \bot \rb \mid \rc` | $\ra \bot \rb \mid \rc$ | They are conditionally independent given $\rc$ | | `P(\ra)` | $P(\ra)$ | A probability distribution over a discrete variable | | `p(\ra)` | $p(\ra)$ | A probability distribution over a continuous variable, or a variable of unspecified type| | `\ra \sim P` | $\ra \sim P$ | Random variable $\ra$ has distribution $P$ | | `\E_{\rx \sim P} [ f(x) ] \text{ or } \E f(x)` | $\E_{\rx \sim P} [ f(x) ] \text{ or } \E f(x)$ | Expectation of $f(x)$ with respect to $P(\rx)$ | | `\Var(f(x))` | $\Var(f(x))$ | Variance of $f(x)$ under $P(\rx)$ | | `\Cov(f(x), g(x))` | $\Cov(f(x), g(x))$ | Covariance of $f(x)$ and $g(x)$ under $P(\rx)$ | | `H(\rx)` | $H(\rx)$ | Shannon entropy of the random variable $\rx$ | | `\KL(P \Vert Q)` | $\KL(P \Vert Q)$ | Kullback-Leibler divergence of $P$ and $Q$ | | `\mathcal{N}(\vx ; \vmu , \mSigma)` | $\mathcal{N}(\vx ; \vmu , \mSigma)$ | Gaussian distribution over $\vx$ with mean $\vmu$ and covariance $\mSigma$ | ### Functions | Command | Visualization | Comment | | --------------------------- | --------------------------- | ------------------------------------------------------------------------------------------------- | | `f: \sA \rightarrow \sB` | $f: \sA \rightarrow \sB$ | The function $f$ with domain $\sA$ and range $\sB$ | | `f \circ g` | $f \circ g$ | Composition of the functions $f$ and $g$ | | `f(\vx ; \vtheta)` | $f(\vx ; \vtheta)$ | A function of $\vx$ parametrized by $\vtheta$. Sometimes written as $f(\vx)$ to simplify notation | | `\log x` | $\log x$ | Natural logarithm of $x$ | | `\sigma(x)` | $\sigma(x)$ | Logistic sigmoid, $\displaystyle \frac{1}{1 + \exp(-x)}$ | | `\zeta(x)` | $\zeta(x)$ | Softplus, $\log(1 + \exp(x))$ | | `\Vert \vx \Vert_p ` | $\Vert \vx \Vert_p$ | $L^p$ norm of $\vx$ | | `\Vert \vx \Vert ` | $\Vert \vx \Vert$ | $L^2$ norm of $\vx$ | | `x^+` | $x^+$ | Positive part of $x$, i.e., $\max(0,x)$ | | `\bm{1}_\mathrm{condition}` | $\bm{1}_\mathrm{condition}$ | Is 1 if the condition is true, 0 otherwise | ### Custom Commands special | Command | Visualization | Comment | | ---------------- | ---------------- | -------------------------------------------- | | `\bm{#1}` | $\bm{x}$ | Bold symbol, e.g., $\boldsymbol{x}$ | | `\sign` | $\sign$ | **operator**, Sign , $\operatorname{sign}$ | | `\Tr` | $\Tr$ | **operator** Trace, $\operatorname{Tr}$ | | `\E` | $\E$ | Expectation, $\mathbb{E}$ | | `\KL` | $\KL$ | Kullback-Leibler divergence, $D_\mathrm{KL}$ | | `\NormalDist` | $\NormalDist$ | Gaussian distribution, $\mathcal{N}$ | | `\diag` | $\diag$ | Diagonal matrix, $\mathrm{diag}$ | | `\Ls` | $\Ls$ | Loss function, $\mathcal{L}$ | | `\R` | $\R$ | Real number set, $\mathbb{R}$ | | `\emp` | $\emp$ | Empirical distribution, $\tilde{p}$ | | `\lr` | $\lr$ | Learning rate, $\alpha$ | | `\reg` | $\reg$ | Regularization coefficient, $\lambda$ | | `\rect` | $\rect$ | Rectifier activation, $\mathrm{rectifier}$ | | `\softmax` | $\softmax$ | Softmax function, $\mathrm{softmax}$ | | `\sigmoid` | $\sigmoid$ | Sigmoid function, $\sigma$ | | `\softplus` | $\softplus$ | Softplus function, $\zeta$ | | `\Var` | $\Var$ | Variance, $\mathrm{Var}$ | | `\standarderror` | $\standarderror$ | Standard error, $\mathrm{SE}$ | | `\Cov` | $\Cov$ | Covariance, $\mathrm{Cov}$ | | `\tran` | $\tran$ | Transpose operator, $^\top$ | | `\inv` | $\inv$ | Inverse operator, $^{-1}$ | | `\diff` | $\diff$ | Differential operator, $\mathrm{d}$ | # Reference - Ian Goodfellow's ML book: https://github.com/goodfeli/dlbook_notation/blob/master/notation_example.pdf - MathJax: https://docs.mathjax.org/en/latest/input/tex/macros.html