Mathematical Notations#

Notation

Definition

\(n\) or \(m\)

The number of observations in a tabular 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 tabular 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).

\(V\)

The vertex set in a graph.

\(E\)

The edge set in a graph.

\(u\), \(v\) or \(w\)

The vertex in a graph.

\((u, v)\)

The edge in a graph.

\(|A|\)

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 the 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.