Computation of sum of squares polynomials from data points

TL;DR

This paper introduces an iterative algorithm to compute sum of squares polynomial approximations from data points, analyzing the convex functional involved and demonstrating its effectiveness for univariate and bivariate cases.

Contribution

It presents a novel convex functional-based iterative method for sum of squares polynomial approximation from data, with theoretical analysis and practical algorithms.

Findings

The functional $G$ is coercive and strictly convex for positive univariate polynomials, ensuring a unique sum of squares decomposition.

For multivariate polynomials with small perturbations, the modified functional $G^ ext{epsilon}$ is coercive, enabling approximate decompositions.

Numerical experiments confirm the effectiveness of the proposed algorithms for univariate and bivariate polynomial data.

Abstract

We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional $G$ arising from the dualization of a quadratic regression over the Cholesky factors of the sum of squares decomposition. In order to justify the construction, the domain of $G$, the boundary of the domain and the behavior at infinity are analyzed in details. When the data interpolate a positive univariate polynomial, we show that in the context of the Lukacs sum of squares representation, $G$ is coercive and strictly convex which yields a unique critical point and a corresponding decomposition in sum of squares. For multivariate polynomials which admit a decomposition in sum of squares and up to a small perturbation of size $\varepsilon$, $G^\varepsilon$ is always coercive and so it minimum yields an approximate decomposition in sum of squares. Various unconstrained descent algorithms are proposed to minimize $G$. Numerical examples are provided, for univariate and bivariate polynomials.

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