Semi-Infinite Linear Regression and Its Applications

TL;DR

This paper introduces semi-infinite linear regression problems involving quasimatrices, providing a formal framework, generalizing algorithms, and proposing sampling methods for solutions, with applications in supervised learning and function approximation.

Contribution

It formalizes semi-infinite linear regression, extends existing algorithms to this setting, and suggests sampling techniques for approximate solutions.

Findings

Semi-infinite linear regression models arise in applications like supervised learning.

Generalized algorithms for finite problems to the semi-infinite case.

Sampling methods can effectively approximate solutions.

Abstract

Finite linear least squares is one of the core problems of numerical linear algebra, with countless applications across science and engineering. Consequently, there is a rich and ongoing literature on algorithms for solving linear least squares problems. In this paper, we explore a variant in which the system's matrix has one infinite dimension (i.e., it is a quasimatrix). We call such problems semi-infinite linear regression problems. As we show, the semi-infinite case arises in several applications, such as supervised learning and function approximation, and allows for novel interpretations of existing algorithms. We explore semi-infinite linear regression rigorously and algorithmically. To that end, we give a formal framework for working with quasimatrices, and generalize several algorithms designed for the finite problem to the infinite case. Finally, we suggest the use of various sampling methods for obtaining an approximate solution.