The masked sample covariance estimator: An analysis using matrix concentration inequalities

Information and Inference

Volumn 1, Issue 1, 2012, Pages 2-20

  Chen, Richard Y ^a   Gittens, Alex ^a   Tropp, Joel A ^a  

^a California Institute of Technology   (United States)

Author keywords

Covariance estimation; Matrix concentration inequality; Matrix Khintchine inequality; Matrix Rosenthal inequality; Random matrix; Schur product

Indexed keywords

EID: 84890096199     PISSN: None     EISSN: 20498772     Source Type: Journal    
DOI: 10.1093/imaiai/ias001     Document Type: Article

References (38)

1. R. Ahlswede A. Winter, 'Strong converse for identification via quantum channels', IEEE Trans. Inform. Theory 48 (2002) 569-579.
2. Z. D. Bai J. W. Silverstein, Spectral analysis of large-dimensional random matrices (Springer, New York, NY, 2010).
3. R. Bhatia, Matrix analysis (Springer, New York, NY, 1997).
4. P. J. Bickel E. Levina, 'Covariance regularization by thresholding', Ann. Statist. 36 (2008) 2577-2604.
5. P. J. Bickel E. Levina, 'Regularized estimation of large covariance matrices', Ann. Statist. 36 (2008) 199-227.
6. A. Buchholz, 'Optimal constants in Khintchine-type inequalities for Fermions, Rademachers and q-Gaussian operators', Bull. Pol. Acad. Sci. Math. 53 (2005) 315-321.
7. T. T. Cai, C.-H. Zhang H. H. Zhou, 'Optimal rates of convergence for covariance matrix estimation', Ann. Statist. 38 (2010) 2118-2144.
8. R. Y. Chen, A. Gittens J. A. Tropp, 'The masked sample covariance estimator: an analysis via the matrix Laplace transform', ACM Report 2012-01, California Institute of Technology, Pasadena, CA.
9. V. H. de la Peña E. Giné, Decoupling: from dependence to independence (Springer, Berlin, 1999).
10. N. El Karoui, 'Operator norm consistent estimation of large dimensional sparse covariance matrices', Ann. Statist. 36 (2008) 2717-2756.
11. D. A. Freedman, Statistical models: theory and practice (Cambridge University Press, Cambridge, 2005).
12. R. Furrer T. Bengtsson, 'Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants', J. Multivariate Anal. 98 (2007) 227-255.
13. D. Gross, 'Recovering low-rank matrices from few coefficients in any basis', IEEE Trans. Inform. Theory 57 (2011) 1548-1566.
14. R. A. Horn C. R. Johnson, Matrix analysis (Cambridge University Press, Cambridge, 1985).
15. R. A. Horn C. R. Johnson, Topics in matrix analysis (Cambridge University Press, Cambridge, 1994).
16. R. A. Johnson D. W. Wichern, Applied multivariate statistical analysis, 6th edn (Prentice Hall, Englewood Cliffs, NJ, 2002).
17. I. T. Jolliffe, Principal component analysis (Springer, New York, NY, 2002).
18. M. Junge Q. Xu, 'Noncommutative Burkholder/Rosenthal inequalities', Ann. Probab. 31 (2003) 948-995.
19. M. Junge Q. Xu, 'On the best constants in some non-commutative martingale inequalities', Bull. London Math. Soc. 37 (2005) 243-253.
20. M. Junge Q. Xu, 'Noncommutative Burkholder/Rosenthal inequalities II: applications', Israel J. Math. 167 (2008) 227-282.
21. M. Junge Q. Zeng, 'Noncommutative Bennett and Rosenthal inequalities', Preprint, 2011, arxiv:1111.1027.
22. M. Ledoux M. Talagrand, Probability in banach spaces: isoperimetry and processes (Springer, Berlin, 1991).
23. E. Levina R. Vershynin, 'Partial estimation of covariance matrices', Probab. Theory Related Fields (2011).
24. p (1 < p < ∞)', C. R. Math. Acad. Sci. Paris 303 (1986) 289-292.
25. F. Lust-Piquard G. Pisier, 'Noncommutative Khintchine and Paley inequalities', Ark. Mat. 29 (1991) 241-260.
26. L. Mackey, M. I. Jordan„ R. Y. Chen, B. Farrell J. A. Tropp, 'Matrix concentration inequalities via the method of exchangeable pairs', Preprint, 2012, arxiv:1201.6002.
27. K. V. Mardia, J. T. Kent J. M. Bibby, Multivariate analysis (Academic Press, London, 1980).
28. R. J. Muirhead, Aspects of multivariate statistical theory (Wiley, New York, NY, 1982).
29. R. I. Oliveira, 'Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges', Preprint, 2010, arxiv:0911.0600.
30. G. Pisier Q. Xu, 'Non-commutative martingale inequalities', Comm. Math. Phys. 189 (1997) 667-698.
31. B. Recht, 'Simpler approach to matrix completion', J. Mach. Learn. Res. 12 (2011) 3413-3430.
32. A. J. Rothman, E. Levina J. Zhu, 'Generalized thresholding of large covariance matrices', J. Amer. Statist. Assoc. 104 (2009) 177-186.
33. M. Rudelson, 'Random vectors in the isotropic position', J. Funct. Anal. 164 (1999) 60-72.
34. M. Rudelson R. Vershynin, 'Sampling from large matrices: an approach through geometric functional analysis', J. Assoc. Comput. Mach. 54 (2007) Article 21, 19 p. (electronic).
35. J. A. Tropp, 'On the conditioning of random subdictionaries', Appl. Comput. Harmon. Anal. 25 (2008) 1-24.
36. J. A. Tropp, 'Freedman's inequality for matrix martingales', Electron. Commun. Probab. 16 (2011) 262-270.
37. J. A. Tropp, 'User-friendly tail bounds for sums of random matrices', Found. Comput. Math. (2011).
38. R. Vershynin, 'Introduction to the non-asymptotic analysis of random matrices', Compressed sensing: theory and applications (eds Y. Eldar and G. Kutyniok; Cambridge University Press, Cambridge, 2011). Available at http://www-personal.umich.edu/ romanv/papers/non-asymptotic-rmt-plain.pdf.