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.. _math_notations:
======================
Mathematical Notations
======================
.. list-table::
:widths: 15 85
:header-rows: 1
* - Notation
- Definition
* - :math:`n` or :math:`m`
- The number of :term:`observations ` in a tabular
:term:`dataset `. Typically :math:`n` is used, but sometimes
:math:`m` is required to distinguish two datasets, e.g., the
:term:`training set ` and the :term:`inference set
`.
* - :math:`p` or :math:`r`
- The number of features in a tabular dataset. Typically :math:`p` is used, but
sometimes :math:`r` is required to distinguish two datasets.
* - :math:`a \times b`
- The dimensionality of a matrix (dataset) has :math:`a` rows
(observations) and :math:`b` columns (features).
* - :math:`V`
- The vertex set in a graph.
* - :math:`E`
- The edge set in a graph.
* - :math:`u`, :math:`v` or :math:`w`
- The vertex in a graph.
* - :math:`(u, v)`
- The edge in a graph.
* - :math:`|A|`
- Depending on the context may be interpreted as follows:
+ If :math:`A` is a set, this denotes its cardinality, i.e., the number
of elements in the set :math:`A`.
+ If :math:`A` is a real number, this denotes the absolute value of
:math:`A`.
* - :math:`\|x\|`
- The :math:`L_2`-norm of a vector :math:`x \in \mathbb{R}^d`,
.. math::
\|x\| = \sqrt{ x_1^2 + x_2^2 + \dots + x_d^2 }.
* - :math:`\mathrm{sgn}(x)`
- Sign function for :math:`x \in \mathbb{R}`,
.. math::
\mathrm{sgn}(x)=\begin{cases}
-1, x < 0,\\
0, x = 0,\\
1, x > 0.
\end{cases}
* - :math:`x_i`
- In the description of an algorithm, this typically denotes the
:math:`i`-th :term:`feature vector ` in the training set.
* - :math:`x'_i`
- In the description of an algorithm, this typically denotes the
:math:`i`-th feature vector in the inference set.
* - :math:`y_i`
- In the description of an algorithm, this typically denotes the
:math:`i`-th :term:`response ` in the training set.
* - :math:`y'_i`
- In the description of an algorithm, this typically denotes the
:math:`i`-th response that needs to be predicted by the inference
algorithm given the feature vector :math:`x'_i` from the inference set.