Equiangular lines via matrix projection

TL;DR

This paper introduces a new matrix projection method to improve bounds on the maximum number of equiangular lines in real space, bridging previous regimes and extending to dense graphs and complex settings.

Contribution

The paper presents a novel approach using orthogonal matrix projection to unify and enhance bounds on equiangular lines, also extending the Alon-Boppana theorem to dense graphs.

Findings

New upper bounds for equiangular lines that are tight or nearly tight.

First extension of the Alon-Boppana theorem to dense graphs.

Application of the method to complex setting scenarios.

Abstract

In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in $\mathbb{R}^r$ with angle $\arccos(\alpha)$ and gave a partial answer in the regime $r \leq 1/\alpha^2 - 2$. At the other extreme where $r$ is at least exponential in $1/\alpha$, recent breakthroughs have led to an almost complete resolution of this problem. In this paper, we introduce a new method for obtaining upper bounds which unifies and improves upon previous approaches, thereby yielding bounds which bridge the gap between the aforementioned regimes and are best possible either exactly or up to a small multiplicative constant. Our approach relies on orthogonal projection of matrices with respect to the Frobenius inner product and as a byproduct, it yields the first extension of the Alon-Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to $\binom{r+1}{2}$ equiangular lines in $\mathbb{R}^r$. Applications of our method in the complex setting will be discussed as well.