The Effective Countable Generalized Moment Problem

TL;DR

This paper develops new convergence rate bounds for the Moment-Sum-of-Squares relaxations of the Generalized Moment Problem with countable constraints, applicable to various GMP instances including tensor decomposition.

Contribution

It introduces geometry-adaptive convergence rates for GMPs under certain conditions, extending previous analyses and applying to tensor decomposition problems.

Findings

Convergence bounds depend on the semi-algebraic set geometry.

Optimizers of relaxations converge to the GMP optimizer under specified conditions.

Provides effective error bounds for tensor decomposition via Moment-SoS hierarchies.

Abstract

We establish new convergence rates for the Moment-Sum-of-Squares (Moment-SoS) relaxations for the Generalized Moment Problem (GMP) with countable moment constraints on vectors of measures, under dual optimum attainment, $S$-fullness and Archimedean conditions. These bounds, which adapt to the geometry of the underlying semi-algebraic set, apply to both the convergence of optima, and to the convergence in Hausdorff distance between the relaxation feasibility set and the GMP feasibility set. We show that under the previous conditions, the sequence of optimizers of the relaxations converge to the optimizer of the GMP for the weak$^*$ topology, provided this optimal measure is unique. This research provides quantitative geometry-adaptive rates for GMPs cast as linear programs on measures. It complements earlier analyses of specific GMP instances (e.g., polynomial optimization) as well as recent methodological frameworks that have been applied to volume computation and optimal control. We apply the convergence rate analysis to symmetric tensor decomposition problems, providing new effective error bounds for the convergence of the Moment-SoS hierarchies for tensor decomposition.