Simplicial faces of the set of correlation matrices

TL;DR

This paper investigates the facial structure of the set of correlation matrices, showing that most small subsets of vertices generate simplicial faces up to a certain size proportional to the square root of the matrix dimension.

Contribution

It establishes a near-optimal bound on the size of vertex sets that generate simplicial faces in the correlation matrix set.

Findings

Almost every set of r vertices generates a simplicial face for r ≤ √(c n)

The maximum size of a simplicial face is on the order of √(2n)

Provides bounds on the facial geometry of correlation matrices

Abstract

This paper concerns the facial geometry of the set of $n \times n$ correlation matrices. The main result states that almost every set of $r$ vertices generates a simplicial face, provided that $r \leq \sqrt{\mathrm{c} n}$, where $\mathrm{c}$ is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than $\sqrt{2n}$ vertices.