Sparse sum of Hermitian squares in group algebras of finite groups

TL;DR

This paper investigates the sum of Hermitian squares in group algebras of finite groups, establishing optimal solutions, proposing a hierarchy, and analyzing exponential decay of residual errors independent of group order in specific cases.

Contribution

It introduces a hierarchy for sparse SOHS problems in finite group algebras and proves exponential decay of residual errors, independent of group order in certain cases.

Findings

Optimal solution of convex relaxation is the square root of the non-negative element.

Residual errors decay exponentially with the hierarchy level.

Decay rate is independent of group order for cyclic or dihedral groups.

Abstract

Non-negative elements in group algebras play a crucial role in the study of functions, measures and operators. This paper focuses on the sum of Hermitian squares (SOHS) of non-negative elements in group algebras of finite groups. We first prove that for a given non-negative element, the optimal solution of the convex relaxation of the sparse SOHS problem is precisely its square root. Then we propose a hierarchy for the sparse SOHS problem, and we analyze the error of the hierarchy with respect to two types of residuals. Notably, we prove that both errors decay exponentially. Moreover, we show that for one type of error, the decay rate is independent of the order of the group. For the other type, we demonstrate that the rate is also independent of the group order, provided that the group is cyclic or dihedral.