Universality in polytope phase transitions and message passing algorithms

Annals of Applied Probability

Volumn 25, Issue 2, 2015, Pages 753-822

  Bayati, Mohsen ^a   Lelarge, Marc ^b   Montanari, Andrea ^c  

^a STANFORD UNIVERSITY   (United States)

^b ECOLE NORMALE SUPÉRIEURE   (France)

^c Stanford University   (United States)

Author keywords

Compressed sensing; Message passing; Polytope neighborliness; Random matrices; Universality

Indexed keywords

EID: 84923415337     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/14-AAP1010     Document Type: Article

References (31)

1. ADAMCZAK, R., LITVAK, A. E., PAJOR, A. and TOMCZAK-JAEGERMANN, N. (2011). Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling. Constr. Approx. 34 61-88. MR2796091
2. AFFENTRANGER, F. and SCHNEIDER, R. (1992). Random projections of regular simplices. Discrete Comput. Geom. 7 219-226. MR1149653
3. ANDERSON, G. W., GUIONNET, A. and ZEITOUNI, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge. MR2760897
4. BAI, Z. and SILVERSTEIN, J. W. (2009). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York. MR2567175
5. BAI, Z. D. and SILVERSTEIN, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316-345. MR1617051
6. BAYATI, M. andMONTANARI, A. (2012). The LASSO risk for Gaussian matrices. IEEE Trans. Inform. Theory 58 1997-2017. MR2951312
7. BOLTHAUSEN, E. (2014). An iterative construction of solutions of the TAP equations for the Sherrington-Kirkpatrick model. Comm. Math. Phys. 325 333-366. MR3147441
8. BÜRGISSER, P. and CUCKER, F. (2010). Smoothed analysis of Moore-Penrose inversion. SIAM J. Matrix Anal. Appl. 31 2769-2783. MR2740632
9. DONOHO, D. and TANNER, J. (2009). Observed universality of phase transitions in highdimensional geometry, with implications for modern data analysis and signal processing. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 4273-4293. MR2546388
10. DONOHO, D. L. (2005). Neighborly polytopes and sparse solution of underdetermined linear equations. Technical report, Statistics Dept., Stanford Univ., Stanford, CA.
11. DONOHO, D. L. (2006). High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35 617-652. MR2225676
12. DONOHO, D. L., JAVANMARD, A. and MONTANARI, A. (2013). Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing. IEEE Trans. Inform. Theory 59 7434-7464. MR3124654
13. DONOHO, D. L., JOHNSTONE, I. andMONTANARI, A. (2013). Accurate prediction of phase transitions in compressed sensing via a connection to minimax denoising. IEEE Trans. Inform. Theory 59 3396-3433. MR3061255
14. DONOHO, D. L.,MALEKI, A. andMONTANARI, A. (2009). Message passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA 106 18914-18919.
15. DONOHO, D. L.,MALEKI, A. andMONTANARI, A. (2011). The noise-sensitivity phase transition in compressed sensing. IEEE Trans. Inform. Theory 57 6920-6941. MR2882271
16. DONOHO, D. L. and TANNER, J. (2005). Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102 9452-9457 (electronic). MR2168716
17. DONOHO, D. L. and TANNER, J. (2005). Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA 102 9446-9451 (electronic). MR2168715
18. DONOHO, D. L. and TANNER, J. (2009). Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Amer. Math. Soc. 22 1-53. MR2449053
19. JAVANMARD, A. and MONTANARI, A. (2013). State evolution for general approximate message passing algorithms, with applications to spatial coupling. Information and Inference 2 115-144.
20. KABASHIMA, Y.,WADAYAMA, T. and TANAKA, T. (2009). A typical reconstruction limit for compressed sensing based on lp-norm minimization. J. Stat. Mech. L09003.
21. KRZAKALA, F., MÉZARD, M., SAUSSET, F., SUN, Y. F. and ZDEBOROVÁ, L. (2012). Statistical-physics-based reconstruction in compressed sensing. Physical Review X 2 021005.
22. LUBINSKY, D. S. (2007). A survey of weighted polynomial approximation with exponential weights. Surv. Approx. Theory 3 1-105. MR2276420
23. MALEKI, A., ANITORI, L., YANG, A. and BARANIUK, R. (2011). Asymptotic analysis of complex LASSO via complex approximate message passing (CAMP). Available at arXiv:1108.0477.
24. MÉZARD, M., PARISI, G. and VIRASORO, M. A. (1987). Spin Glass Theory and Beyond. World Scientific Lecture Notes in Physics 9. World Scientific, Teaneck, NJ. MR1026102
25. RANGAN, S. (2011). Generalized approximate message passing for estimation with random linear mixing. In IEEE Intl. Symp. on Inform. Theory (St. Perersbourg), August 2011. IEEE, Piscataway, NJ.
26. RANGAN, S., FLETCHER, A. K. and GOYAL, V. K. (2009). Asymptotic analysis of map estimation via the replica method and applications to compressed sensing. In Neural Information Processing Systems (NIPS). Vancouver.
27. SCHNITER, P. (2010). Turbo reconstruction of structured sparse signals. In Proceedings of the Conference on Information Sciences and Systems. Princeton, NJ.
28. SCHNITER, P. and RANGAN, S. (2012). Compressive phase retrieval via generalized approximate message passing. In Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on 815-822. IEEE, Piscataway, NJ.
29. TAO, T. and VU, V. (2012). Random matrices: The Universality phenomenon for Wigner ensembles. Available at arXiv:1202.0068.
30. THOULESS, D. J.,ANDERSON, P.W. and PALMER, R. G. (1977). Solution of "solvable model of a spin glass." Philosophical Magazine 35 593-601.
31. VERSHIK, A. M. and SPORYSHEV, P. V. (1992). Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem. Selecta Math. Soviet. 11 181-201. Selected translations. MR1166627