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

## TL;DR

This paper explores the connection between equiangular tight frames and sum-of-squares optimization, deriving new bounds on ETF sparsity and highlighting optimal sparsity in Steiner ETFs related to finite projective planes.

## Contribution

The authors generalize previous calculations to establish new bounds on the sparsity of real and complex ETFs, and identify Steiner ETFs as optimally sparse in certain matrix inequalities.

## Key findings

- Steiner ETFs achieve tight bounds on sparsity.
- New bounds on ETF sparsity are derived.
- Open problems for further generalizations are proposed.

## 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.