Sampling algebraic varieties for sum of squares programs

TL;DR

This paper introduces a sampling-based approach for sum of squares relaxations over algebraic varieties, leveraging geometric sampling to simplify SDPs especially for complex varieties with easy sampling methods.

Contribution

It proposes a novel sampling methodology that bypasses algebraic descriptions, reducing SDP size and connecting SOS optimization with numerical algebraic geometry.

Findings

Effective for varieties with easy sampling but complex defining equations

Reduces SDP size by exploiting coordinate ring structure

Applicable to varieties like $SO(n)$, Grassmannians, and rank $k$ tensors

Abstract

We study sum of squares (SOS) relaxations to optimize polynomial functions over a set $V\cap R^n$, where $V$ is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents $V$ with a generic set of complex samples. This approach depends only on the geometry of $V$, avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the coordinate ring structure to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties that are easy to sample from but for which the defining equations are complicated, such as $SO(n)$, Grassmannians or rank $k$ tensors. For arbitrary varieties we can obtain the required samples by using the tools of numerical algebraic geometry. In this way we connect the areas of SOS optimization and numerical algebraic geometry.