# Mathematical Notations#

Notation

Definition

$$n$$ or $$m$$

The number of observations in a dataset. Typically $$n$$ is used, but sometimes $$m$$ is required to distinguish two datasets, e.g., the training set and the inference set.

$$p$$ or $$r$$

The number of features in a dataset. Typically $$p$$ is used, but sometimes $$r$$ is required to distinguish two datasets.

$$a \times b$$

The dimensionality of a matrix (dataset) has $$a$$ rows (observations) and $$b$$ columns (features).

$$|A|$$

Depending on the context may be interpreted as follows:

• If $$A$$ is a set, this denotes its cardinality, i.e., the number of elements in the set $$A$$.

• If $$A$$ is a real number, this denotes an absolute value of $$A$$.

$$\|x\|$$

The $$L_2$$-norm of a vector $$x \in \mathbb{R}^d$$,

$\|x\| = \sqrt{ x_1^2 + x_2^2 + \dots + x_d^2 }.$

$$\mathrm{sgn}(x)$$

Sign function for $$x \in \mathbb{R}$$,

$\begin{split}\mathrm{sgn}(x)=\begin{cases} -1, x < 0,\\ 0, x = 0,\\ 1, x > 0. \end{cases}\end{split}$

$$x_i$$

In the description of an algorithm, this typically denotes the $$i$$-th feature vector in the training set.

$$x'_i$$

In the description of an algorithm, this typically denotes the $$i$$-th feature vector in the inference set.

$$y_i$$

In the description of an algorithm, this typically denotes the $$i$$-th response in the training set.

$$y'_i$$

In the description of an algorithm, this typically denotes the $$i$$-th response that needs to be predicted by the inference algorithm given the feature vector $$x'_i$$ from the inference set.