Polynomial argmin for recovery and approximation of multivariate discontinuous functions

TL;DR

This paper introduces a polynomial-based, mesh-free method for approximating multivariate functions, including discontinuous ones, using semidefinite programming, which is efficient and does not require prior knowledge of discontinuities.

Contribution

It presents a novel polynomial argmin approach that is mesh-free, model-free, and scalable, with theoretical guarantees for approximation accuracy and sample complexity.

Findings

Exact approximation for piecewise polynomial functions with low-degree polynomials

Semidefinite program size is independent of ambient dimension

Provides probabilistic generalization error bounds

Abstract

We propose to approximate a (possibly discontinuous) multivariate function f (x) on a compact set by the partial minimizer arg miny p(x, y) of an appropriate polynomial p whose construction can be cast in a univariate sum of squares (SOS) framework, resulting in a highly structured convex semidefinite program. In a number of non-trivial cases (e.g. when f is a piecewise polynomial) we prove that the approximation is exact with a low-degree polynomial p. Our approach has three distinguishing features: (i) It is mesh-free and does not require the knowledge of the discontinuity locations. (ii) It is model-free in the sense that we only assume that the function to be approximated is available through samples (point evaluations). (iii) The size of the semidefinite program is independent of the ambient dimension and depends linearly on the number of samples. We also analyze the sample complexity of the approach, proving a generalization error bound in a probabilistic setting. This allows for a comparison with machine learning approaches.