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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.

References

  1. Ahlswede, R., Winter, A.: Strong converse for identification via quantum channels. IEEE Trans. Inform. Theory 48(3), 569–579 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS/SIAM Series on Optimization. SIAM, Philadelphia, PA (2001)

    Book  MATH  Google Scholar 

  3. Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1997)

    MATH  Google Scholar 

  4. Delorme, C., Poljak, S.: The performance of an eigenvalue bound on the max-cut problem in some classes of graphs. Discrete Math. 111(1–3), 145–156 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8, 1–79 (1953)

    MathSciNet  MATH  Google Scholar 

  7. Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  8. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)

    Chapter  Google Scholar 

  9. Khot, S., Naor, A.: Grothendieck-type inequalities in combinatorial optimization. Commun. Pure Appl. Math. 65(7), 992–1035 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koltchinskii, V., Mendelson, S.: Bounding the smallest singular value of a random matrix without concentration. Int. Math. Res. Not. IMRN 2015(23), 12991–13008 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Laurent, M., Poljak, S.: On a positive semidefinite relaxation of the cut polytope. Linear Algebra Appl. 223(224), 439–461 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Laurent, M., Poljak, S.: On the facial structure of the set of correlation matrices. SIAM J. Matrix Anal. Appl. 17(3), 530–547 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. McCoy, M., Tropp, J.A.: Two proposals for robust PCA using semidefinite programming. Electron. J. Stat. 5, 1123–1160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nesterov, Yu.: Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9(1–3), 141–160 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, New York (2014)

    Book  MATH  Google Scholar 

  16. Oliveira, R.I.: The lower tail of random quadratic forms with applications to ordinary least squares. Probab. Theory Relat. Fields 166(3–4), 1175–1194 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. 49(2), 237–323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Poljak, S., Rendl, F.: Solving the max-cut problem using eigenvalues. Discrete Appl. Math. 62(1–3), 249–278 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton, NJ (1970)

    Book  MATH  Google Scholar 

  20. Tropp, J.A.: User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12(4), 389–434 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tropp, J.A.: Convex recovery of a structured signal from independent random linear measurements. In: Pfander, G.E. (ed.) Sampling Theory, a Renaissance. Applied and Numerical Harmonic Analysis, pp. 67–101. Birkhäuser, Cham (2015)

    Chapter  Google Scholar 

  22. Tropp, J.A.: An introduction to matrix concentration inequalities. Found. Trends Mach. Learn. 8(1–2), 1–230 (2015)

    Article  MATH  Google Scholar 

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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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