VectorSpaceLeastSquares
This is the documentation for release 1.1.0
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
- The sequence $(x_m, y_m)_{1 \le m \le M}$ is the training data: $y_m \in \mathbb{R}$ is the expected value at point $x_m \in \mathbb{R}^d$.
- The function $\varphi: \mathbb{R}^d \to \mathbb{R}^d$ is the transformation to be be applied to the input data before solving the least squares problem
- For $i = 1, \dots, d$, $g_i : \mathbb{R}^d \to \mathbb{R}$. The family $(g_1, \dots, g_d)$ must be a free family to ensure the least squares problem is not ill-posed. 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 $\varphi$.
Installation
VectorSpaceLeastSquares is compatible with Julia version 1.11 or later.
To get the VectorSpaceLeastSquares package installed, execute the following Julia command:
import Pkg
Pkg.add("VectorSpaceLeastSquares")