# Notation¶

Throughout this book, we adhere to the following notational conventions. Note that some of these symbols are placeholders, while others refer to specific objects. As a general rule of thumb, the indefinite article “a” often indicates that the symbol is a placeholder and that similarly formatted symbols can denote other objects of the same type. For example, “$$x$$: a scalar” means that lowercased letters generally represent scalar values, but “$$\mathbb{Z}$$: the set of integers” refers specifically to the symbol $$\mathbb{Z}$$.

## Numerical Objects¶

• $$x$$: a scalar

• $$\mathbf{x}$$: a vector

• $$\mathbf{X}$$: a matrix

• $$\mathsf{X}$$: a general tensor

• $$\mathbf{I}$$: the identity matrix (of some given dimension), i.e., a square matrix with $$1$$ on all diagonal entries and $$0$$ on all off-diagonals

• $$x_i$$, $$[\mathbf{x}]_i$$: the $$i^\textrm{th}$$ element of vector $$\mathbf{x}$$

• $$x_{ij}$$, $$x_{i,j}$$,$$[\mathbf{X}]_{ij}$$, $$[\mathbf{X}]_{i,j}$$: the element of matrix $$\mathbf{X}$$ at row $$i$$ and column $$j$$.

## Set Theory¶

• $$\mathcal{X}$$: a set

• $$\mathbb{Z}$$: the set of integers

• $$\mathbb{Z}^+$$: the set of positive 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

• $$|\mathcal{X}|$$: cardinality (number of elements) of set $$\mathcal{X}$$

• $$\mathcal{A}\cup\mathcal{B}$$: union of sets $$\mathcal{A}$$ and $$\mathcal{B}$$

• $$\mathcal{A}\cap\mathcal{B}$$: intersection of sets $$\mathcal{A}$$ and $$\mathcal{B}$$

• $$\mathcal{A}\setminus\mathcal{B}$$: set subtraction of $$\mathcal{B}$$ from $$\mathcal{A}$$ (contains only those elements of $$\mathcal{A}$$ that do not belong to $$\mathcal{B}$$)

## Functions and Operators¶

• $$f(\cdot)$$: a function

• $$\log(\cdot)$$: the natural logarithm (base $$e$$)

• $$\log_2(\cdot)$$: logarithm to base $$2$$

• $$\exp(\cdot)$$: the exponential function

• $$\mathbf{1}(\cdot)$$: the indicator function; evaluates to $$1$$ if the boolean argument is true, and $$0$$ otherwise

• $$\mathbf{1}_{\mathcal{X}}(z)$$: the set-membership indicator function; evaluates to $$1$$ if the element $$z$$ belongs to the set $$\mathcal{X}$$ and $$0$$ otherwise

• $$\mathbf{(\cdot)}^\top$$: transpose of a vector or a matrix

• $$\mathbf{X}^{-1}$$: inverse of matrix $$\mathbf{X}$$

• $$\odot$$: Hadamard (elementwise) product

• $$[\cdot, \cdot]$$: concatenation

• $$\|\cdot\|_p$$: $$\ell_p$$ norm

• $$\|\cdot\|$$: $$\ell_2$$ norm

• $$\langle \mathbf{x}, \mathbf{y} \rangle$$: inner (dot) product of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$

• $$\sum$$: summation over a collection of elements

• $$\prod$$: product over a collection of elements

• $$\stackrel{\textrm{def}}{=}$$: an equality asserted as a definition of the symbol on the left-hand side

## 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¶

• $$X$$: a random variable

• $$P$$: a probability distribution

• $$X \sim P$$: the random variable $$X$$ follows distribution $$P$$

• $$P(X=x)$$: the probability assigned to the event where random variable $$X$$ takes value $$x$$

• $$P(X \mid Y)$$: the conditional probability distribution of $$X$$ given $$Y$$

• $$p(\cdot)$$: a probability density function (PDF) associated with distribution $$P$$

• $${E}[X]$$: expectation of a random variable $$X$$

• $$X \perp Y$$: random variables $$X$$ and $$Y$$ are independent

• $$X \perp Y \mid Z$$: random variables $$X$$ and $$Y$$ are conditionally independent given $$Z$$

• $$\sigma_X$$: standard deviation of random variable $$X$$

• $$\textrm{Var}(X)$$: variance of random variable $$X$$, equal to $$\sigma^2_X$$

• $$\textrm{Cov}(X, Y)$$: covariance of random variables $$X$$ and $$Y$$

• $$\rho(X, Y)$$: the Pearson correlation coefficient between $$X$$ and $$Y$$, equals $$\frac{\textrm{Cov}(X, Y)}{\sigma_X \sigma_Y}$$

• $$H(X)$$: entropy of random variable $$X$$

• $$D_{\textrm{KL}}(P\|Q)$$: the KL-divergence (or relative entropy) from distribution $$Q$$ to distribution $$P$$

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