.. ****************************************************************************** .. * Copyright 2019-2021 Intel Corporation .. * .. * Licensed under the Apache License, Version 2.0 (the "License"); .. * you may not use this file except in compliance with the License. .. * You may obtain a copy of the License at .. * .. * http://www.apache.org/licenses/LICENSE-2.0 .. * .. * Unless required by applicable law or agreed to in writing, software .. * distributed under the License is distributed on an "AS IS" BASIS, .. * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. .. * See the License for the specific language governing permissions and .. * limitations under the License. .. *******************************************************************************/ .. _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.