The outliers among the singular values of large rectangular random matrices with additive fixed rank deformation

Markov Processes and Related Fields

Volumn 20, Issue 2, 2014, Pages 183-228

  Chapon, F ^a   Couillet, R ^b   Hachem, W ^c   Mestre, X ^d  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b SUPELEC   (France)

^c CNRS   (France)

^d CENTRE TECNOLÒGIC DE TELECOMUNICACIONS DE CATALUNYA CTTC   (Spain)

Author keywords

Extreme eigenvalues; Fixed rank deformation; Gaussian fluctuations; Random matrix theory; Stieltjes transform

Indexed keywords

EID: 84930623776     PISSN: 10242953     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (44)

1. Z. BAI AND J.-F. YAO (2008) Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. H. Poincaré, Probabilités et Statistiques 44 (3), 447-474.
2. Z.D. BAI (1999) Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 (3), 611-677.
3. Z.D. BAI AND J.W. SILVERSTEIN (1998) No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Prob. 26 (1), 316-345.
4. Z.D. BAI AND J.W. SILVERSTEIN (1999) Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Prob. 27 (3), 1536-1555.
5. [] J. BAIK, G. BEN AROUS AND S. PÉCHÉ (2005) Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Prob. 33 (5), 1643-1697.
6. J. BAIK AND J.W. SILVERSTEIN (2006) Eigenvalues of lange sample covariance matrices of spiked population models. J. Multivar. Anal. 97 (6), 1382-1408.
7. F. BENAYCH-GEORGES, A. GUIONNET AND M. MA'IDA (2011) Fluctuations of the extreme eigenvadues of finite rank deformations of randam matrices. Electron. J. Prob. 16, 1621-1662. Paper no. 60.
8. F. BENAYCH-GEORGES AND R.R. NADAKUDITI (2011) The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math. 227 (1), 494-521.
9. F. BENAYCH-GEORGES AND R.R. NADAKUDITI (2012) The singular values and vectors of low rank perturbations of large rectangular random matrices. J. Multivar. Anal. 111, 295-309.
10. V. BOLOTNIKOV (1997) On a general moment problem on the half axis. Linear Algebra Appl. 255, 57-112.
11. M. CAPITAINE, C. DONATI-MARTIN AND D. FÉRAL (2009) The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations. Ann. Prob. 37 (1), 1-47.
12. M. CAPITAINE, C. DONATI-MARTIN AND D. FERAL (2012) Central limit theorems for eigenvalues of deformations of Wigner matrices. Ann. Inst. H. Poincaré, Probabilités et Statistiques 48 (1), 107-133.
13. L.H.Y. CHEN (1982) An inequality for the multivariate normal distribution. J. Multivar. Anal. 12 (2), 306-315.
14. R. COUILLET AND W. HACHEM (2013) Fluctuations of spiked random matrix models and failure diagnosis in sensor networks. IEEE Trans. Inform. Theory. 59 (1), 509-525.
15. F. GESZTESY AND E. TSEKANOVSKII (2000) On matrix-valued Herglotz functions. Math. Nachr. 218, 61-138.
16. [] V.L. GIRKO (1990) Theory of Random Determinants. Mathematics and its Applications (Soviet Series) 45, Kluwer Academic Publishers Group, Dordrecht.
17. U. HAAGERUP AND S. THORBJØRNSEN (2005) A new application of random matrices: Ext(C*ed(F2)) is not a group. Ann. Math. 162 (2), 711-775.
18. [] W. HACHEM, O. KHORUNZHY, P. LOUBATON, J. NAJÍM AND L. PASTUR (2008) A new approach for mutual information analysis of large dimensional multi-antenna channels. IEEE Trans. Inform. Theory 54 (9), 3987-4004.
19. W. HACHEM, P. LOUBATON, X. MESTRE, J. NAJIM AND P. VALLET (2012) Large information plus noise random matrix models and consistent subspace estimation in large sensor networks. Random Matrices: Theor. Appl. 1 (2). arXiv:1106.5119.
20. W. HACHEM, P. LOUBATON, X. MESTRE, J. NAJÍM AND P. VALLET (2013) A subspace estimator for fixed rank perturbations of large random matrices. J. Multivar. Anal. 114, 427-447.
21. W. HACHEM, P. LOUBATON, J. NAJÍM AND P. VALLET (2013) On bilinear forms based on the resolvent of large random matrices. Ann. Inst. H. Poincaré, Probabilités et Statistiques 49 (1), 36-63.
22. W. HACHEM, PH. LOUBATON AND J. NAJIM (2007) Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 (3), 875-930.
23. I. M. JOHNSTONE (2001) On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2), 295-327.
24. A. M. KHORUNZHY AND L. A. PASTUR (1993) Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness. Commun. Math. Phys. 153 (3), 605-646.
25. A. KNOWLES AND J. YIN (2013) The isotropic semicircle law and deformation of Wigner matrices. Comm. Pure and Appl. Math. 66 (11), 1663-1749.
26. A. KNOWLES AND J. YIN (2012) The outliers of a deformed Wigner matrix. To appear in Ann. Prob. arXiv: 1207.5619.
27. M. G. KREǏN AND A. A. NUDEL'MAN (1977) The Markov Moment Problem and Extremal Problems. Ideas and problems of P.L. Čebyšev and A.A. Markov and their further development, Translated from the Russian by D. Louvish, Translations of Mathematical Monographs 50, American Mathematical Society, Providence, R.I.
28. V. A. MARČENKO AND L.A. PASTUR (1967) Distribution of eigenvalues in certain sets of random matrices. Mat. Sbornik (N.S.) 72 (114), 507-536.
29. X. MESTRE (2008) Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Trans. Inform. Theory 54 (11), 5113-5129.
30. X. MESTRE (2008) On the asymptotic behavior of the sample estimates of eigen-values and eigenvectors of covariance matrices. IEEE Trans. Signal Process. 56 (11), 5353-5368.
31. T. NÆS, T. ISAKSSON, T. FEARN AND T. DAVIES (2002) A User-Friendly Guide to Multivariate Calibration and Classification. NIR Publications, Chichester, UK.
32. L.A. PASTUR, A.M. KHORUNZHIǏ AND V.YU. VASIL'CHUK (1995) On an asymptotic property of the spectrum of the sum of one-dimensional independent random operators. Dopov. Nats. Akad. Nauk Ukraïni 2, 27-30.
33. L.A. PASTUR AND M. SHCHERBINA (2011) Eigenvalue Distribution of Large Random Matrices. Mathematical Surveys and Monographs 171, American Mathematical Society, Providence, RI.
34. [] D. PAUL (2007) Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 (4), 1617-1642.
35. S. PÉCHÉ (2006) The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory and Relat. Fields 134 (1), 127-173.
36. A. Pizzo, D. RENFREW AND A. SOSHNIKOV (2013) On finite rank deformations of Wigner matrices. Ann. Inst. H. Poincaré, Probabilités et Statistiques 49 (1), 64-94.
37. M. POTTERS, J.-P. BOUCHAUD AND L. LALOUX (2005) Financial applications of random matrix theory: Old laces and new pieces. arXiv : physics/0507111.
38. D. RENFREW AND A. SOSHNIKOV (2013) On finite rank deformations of Wigner matrices II: delocalized perturbations. Random Matrices: Theor. Appl. 2 (1), 36 pp.
39. J.W. SILVERSTEIN (1995) Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivar. Anal. 55 (2), 331-339.
40. J.W. SILVERSTEIN AND Z.D. BAI (1995) On the empirical distribution of eigen-values of a class of large-dimensional random matrices. J. Multivar. Anal. 54 (2), 175-192.
41. J.W. SILVERSTEiN AND S.-IL CHOI (1995) Analysis of the limiting spectral distribution of large-dimensional random matrices. J. Multivar. Anal. 54 (2), 295-309.
42. D. TSE AND P. VISWANATH (2005) Fundamentals of Wireless Communication. Cambridge university press.
43. P. VALLET, P. LOUBATON AND X. MESTRE (2012) Improved subspace estimation for multivariate observations of high dimension: The deterministic signals case. IEEE Trans. Inform. Theory 58 (2), 1043 -1068.
44. J. VINOGRADOVA, R. COUILLET AND W. HACHEM (2013) Statistical inference in large antenna arrays under unknown noise pattern. IEEE Trans. Signal Process. 61 (22), 5633-5645.