VectorSpaceLeastSquares

This package aims at computing the least squares approximation within a vector space of functions. It solves the following optimization problem

$$\inf_{\alpha \in \mathbb{R}^d} \sum_{m=1}^M \left(\sum_{i=1}^d \alpha_i g_i\circ\varphi(x_m) - y_m\right)^2$$

where

• $(x_m, y_m)_{1 \le m \le M}$ are training values: $y_m \in \mathbb{R}$ is the expected value at point $x_m \in \mathbb{R}^d$.
• $\varphi: \mathbb{R}^d \to \mathbb{R}^d$ is a transformation to be be applied to the input data before solving the least squares problem
• For $i = 1, \dots, d$, $g_i : \mathbb{R}^d \to \mathbb{R}$ and the family $(g_1, \dots, g_d)$ is a free family. We call the family a basis in the following.

Let $\mathcal{V}$ be the vector space generated by the functions $(g_1, \dots, g_d)$, this package computes the best least squares approximation of the unknown function $x \longmapsto y$ inside $\mathcal{V}$ up to a transformation $\varphi$.

The complete manual is available at https://jlelong.github.io/VectorSpaceLeastSquares.jl.