Optimal data fitting: a moment approach

TL;DR

This paper introduces a moment relaxation method for solving separation and covering problems involving semi-algebraic sets, demonstrating finite convergence and providing a practical algorithm with promising computational results.

Contribution

The paper develops a novel moment relaxation approach for polynomial-based separation and covering problems, showing finite convergence and offering a scalable iterative algorithm.

Findings

Optimal relaxation value converges finitely as moment order increases.

Perturbed problems achieve convergence at degree d.

Algorithm performs well on large datasets.

Abstract

We propose a moment relaxation for two problems, the separation and covering problem with semi-algebraic sets generated by a polynomial of degree d. We show that (a) the optimal value of the relaxation finitely converges to the optimal value of the original problem, when the moment order r increases and (b) after performing some small perturbation of the original problem, convergence can be achieved with r=d. We further provide a practical iterative algorithm that is computationally tractable for large datasets and present encouraging computational results.