Simplicial faces of the set of correlation matrices
Joel A. Tropp

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 correlation matrices. The main result states that almost every set of vertices generates a simplicial face, provided that , where is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than vertices.
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