Sum-of-Squares Optimization and the Sparsity Structure of Equiangular Tight Frames

Abstract

Equiangular tight frames (ETFs) may be used to construct examples of feasible points for semidefinite programs arising in sum-of-squares (SOS) optimization. We show how generalizing the calculations in a recent work of the authors' that explored this connection also yields new bounds on the sparsity of (both real and complex) ETFs. One corollary shows that Steiner ETFs corresponding to finite projective planes are optimally sparse in the sense of achieving tightness in a matrix inequality controlling overlaps between sparsity patterns of distinct rows of the synthesis matrix. We also formulate several natural open problems concerning further generalizations of our technique.The v i form an equiangular tight frame (ETF) if moreover there exists α ∈ [0, 1] such thatWhen moreover v i ∈ R r , we have X ∈ E N , and such points form an interesting subset of the elliptope's boundary. In both the real and complex cases, UNTFs and ETFs have been studied in great detail previously (e.g. [3,4,27,5,15]).The set E N , however, is only the first of a sequence of tightening relaxations of C N , described by the sum-of-squares (SOS) hierarchy. This gives sets we call degree d generalized elliptopes E N d for positive even integers d.where, identifying indices of Y with elements of [N ] d/2 , the following conditions hold.