Second order conditions to decompose smooth functions as sums of squares

TL;DR

This paper establishes second order sufficient conditions for representing non-negative smooth functions as sums of squares of differentiable functions, extending previous results to functions with zero sets that are continuous or manifolds, with applications in optimization.

Contribution

It provides new second order conditions guaranteeing regularity in sum of squares decompositions for functions with continuous zero sets or manifolds, broadening applicability.

Findings

Conditions apply to functions with zero sets on manifolds

Ensures regularity of sum of squares decompositions

Applicable to optimization problems on manifolds

Abstract

We consider the problem of decomposing a regular non-negative function as a sum of squares of functions which preserve some form of regularity. In the same way as decomposing non-negative polynomials as sum of squares of polynomials allows to derive methods in order to solve global optimization problems on polynomials, decomposing a regular function as a sum of squares allows to derive methods to solve global optimization problems on more general functions. As the regularity of the functions in the sum of squares decomposition is a key indicator in analyzing the convergence and speed of convergence of optimization methods, it is important to have theoretical results guaranteeing such a regularity. In this work, we show second order sufficient conditions in order for a $p$ times continuously differentiable non-negative function to be a sum of squares of $p-2$ differentiable functions. The main hypothesis is that, locally, the function grows quadratically in directions which are orthogonal to its set of zeros. The novelty of this result, compared to previous works is that it allows sets of zeros which are continuous as opposed to discrete, and also applies to manifolds as opposed to open sets of $\R^d$. This has applications in problems where manifolds of minimizers or zeros typically appear, such as in optimal transport, and for minimizing functions defined on manifolds.