Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets

TL;DR

This paper analyzes the convergence rates of moment-sum-of-squares hierarchies used for approximating the volume of semialgebraic sets, establishing a lower bound on the asymptotic convergence rate.

Contribution

It provides the first theoretical analysis of the convergence rate of these hierarchies, showing it is at least O(1/ log log d).

Findings

Convergence rate is at least O(1/ log log d).

Hierarchies approximate the indicator function from above.

Provides theoretical bounds for volume approximation methods.

Abstract

Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set K. The idea consists of approximating from above the indicator function of K with a sequence of polynomials of increasing degree d, so that the integrals of these polynomials generate a convergence sequence of upper bounds on the volume of K. We show that the asymptotic rate of this convergence is at least O(1/ log log d).