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Pavel Erofeev edited GPR.tex
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\section{Gaussian Processes Regression}
\label{sec:GaussianProcessesRegression}
In multidimensional regression problem we assume that $f: \mathbb{X} \rightarrow \mathbb{R}, \mathbb{X}\subset\mathbb{R}^m$ is an unknow dependency function. We are given with a so-called \textit{learning set}
$\mathcal{D} $$\mathcal{D} = \bigl(X, Y\bigr) = \bigl\{\bigl(\mathbf{x}_i, y_i = f(\mathbf{x}_i)\bigr), i = \overline{1, N}\bigr,
\mathbf{x]}\in\mathbb{X}\}$. \mathbf{x]}\in\mathbb{X}\}$$. The problem is to construct function $\hat{f}$ from specific class $(\vecX) = \hat{f}(\vecX | D_{learn})$ для исходной зависимости $\vecY = f(\vecX)$ по обучающей выборке $D_{learn}$.
Если для всех $\vecX \in \XX$ (не только для $\vecX \in D_{learn}$) имеет место примерное равенство
\begin{equation}\label{eq:good_approx}