Simplicial Faces of the Set of Correlation Matrices

Abstract

This paper concerns the facial geometry of the set of n×n correlation matrices. The main result states that almost every set of r vertices generates a simplicial face, provided that r ≤ √cn, where c is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than √2n vertices.

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© Springer Science+Business Media, LLC, part of Springer Nature 2017. Received: 5 December 2016 / Revised: 10 November 2017 / Accepted: 15 December 2017. The author thanks Richard Küng and Benjamin Recht for helpful conversations related to this work. This research was partially supported by ONR award N00014-11-1002 and the Gordon & Betty Moore Foundation.

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