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Appendix A. Mathematics Terminology Reference

To serve as either a refresher or an introduction, Table A-1 summarizes the key mathematical terminology in the context in which it is used in this book.

Table A-1. Mathematics terminology summary
Terminology	Description
x	A nonbold variable refers to a scalar.
x	A variable in bold refers to a vector.
y = f(x; Θ)	The result of function f on the vector input x, where f is dependent on the parameters Θ. In the context of this book, this represents the output of a DNN model for a particular input: f represents the DNN model algorithm, Θ represents its parameters determined during training, and x is the input to the model.
C(f(x; Θ), y)	The result of function C given f(x; Θ) and the vector y. In the context of this book, this represents the cost (or loss) of the DNN model for a particular input with respect to the required output y.
x_i	The element i of vector x.
$\frac{d y}{d x}$ $\frac{d y}{d x}$	The derivative of y with respect to x.
$\frac{\partial y}{\partial x}$ $\frac{\partial y}{\partial x}$	The partial derivative of y with respect to x, wh - ere x is one of the variables that affects y.
$\nabla_{x} f$ $\nabla_{x} f$	The nabla (upside-down Greek delta) symbol means “gradient.” $\nabla_{x} f$ $\nabla_{x} f$ refers to the vector of partial derivatives of the function f for the vector x. Put more simply, this means the effect that a very small change to the value of x has on the function f.
{1, 2, . . . L}	The set of numbers from 1 to L.
ℝ	The set of real numbers.
{x : P(x)}	The set of all values of x for which P(x) is true, where P(x) is a boolean statement.
$\underset{x}{arg min} {f (x) : P (x)}$ $\underset{x}{arg min} {f (x) : P (x)}$	Returns x at which f(x) is minimized such that P(x) is true.
x ∈ ℝ	Indicates that x belongs to the set of real numbers.
${\| \| x \| \|}_{p}$ ${\| \| x \| \|}_{p}$	The L^p-norm of the vector x. L^p-norms are explained in “A Mathematical Approach to Measuring Perturbation”.
∑ x	The sum of all the possible values of x.
$\sum_{x \neq t} x$ $\sum_{x \neq t} x$	The sum of all the possible values of x where $x \neq t$ $x \neq t$ .
ε	The Greek letter ε (epsilon) is used to indicate an extremely small (infinitesimal) quantity.

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Table of Contents for A. Mathematics Terminology Reference

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Appendix A. Mathematics Terminology Reference

Table of Contents for
A. Mathematics Terminology Reference