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Simplicial Faces of the Set of Correlation Matrices

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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 \(r \le \sqrt{\mathrm {c} n}\), 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.

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Notes

  1. In our parameter regime, it is unlikely that any vertex of \(\mathscr {E}_n\) is chosen more than once, so this model is not substantially different from drawing vertices without replacement.

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Acknowledgements

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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  1. California Institute of Technology, Pasadena, CA, 91125-5000, USA

    Joel A. Tropp

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  1. Joel A. Tropp
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Correspondence to Joel A. Tropp.

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Tropp, J.A. Simplicial Faces of the Set of Correlation Matrices. Discrete Comput Geom 60, 512–529 (2018). https://doi.org/10.1007/s00454-017-9961-0

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