Sparse sum of Hermitian squares in group algebras of finite groups

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.