Sparse Approximate Solutions to Max-Plus Equations with Application to Multivariate Convex Regression

TL;DR

This paper introduces an efficient polynomial-time method for finding sparse approximate solutions to max-plus equations and applies it to convex multivariate function fitting with guarantees on model optimality and minimal complexity.

Contribution

It presents a novel polynomial-time approach for sparse solutions to max-plus equations and applies it to convex regression with optimality and minimal region guarantees.

Findings

Efficient polynomial-time algorithms for sparse max-plus solutions.

Application to convex multivariate function fitting with optimality guarantees.

Achieves approximately minimal number of affine regions in the model.

Abstract

In this work, we study the problem of finding approximate, with minimum support set, solutions to matrix max-plus equations, which we call sparse approximate solutions. We show how one can obtain such solutions efficiently and in polynomial time for any $\ell_p$ approximation error. Based on these results, we propose a novel method for piecewise-linear fitting of convex multivariate functions, with optimality guarantees for the model parameters and an approximately minimum number of affine regions.