1-Bit matrix completion

Information and Inference

Volumn 3, Issue 3, 2014, Pages 189-223

  Davenport, Mark A ^a   Plan, Yaniv ^b   Van Den Berg, Ewout ^c   Wootters, Mary ^d  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^c IBM T J WATSON RESEARCH CENTER   (United States)

^d UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Binary observations; Constrained maximum likelihood estimation; Convex programming; Matrix completion; Quantization

Indexed keywords

EID: 84929164125     PISSN: None     EISSN: 20498772     Source Type: Journal    
DOI: 10.1093/imaiai/iau006     Document Type: Article

References (68)

1. Ai, A., Lapanowski, A., Plan, Y. & Vershynin, R. (2014) One-bit compressed sensing with non-gaussian measurements. Linear Alg. Appl., 441, 222-239.
2. Bach, F. (2010) Self-concordant analysis for logistic regression. Elecctron. J. Stat., 4, 384-414.
3. Barzilai, J. & Borwein, J. (1988) Two-point step size gradient methods. IMA J. Num. Anal., 8, 141-148.
4. Beck, A. & Teboulle, M. (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci., 2, 183-202.
5. Becker, S., Candès, E. & Grant, M. (2011) Templates for convex cone problems with applications to sparse signal recovery. Math. Program. Comput., 3, 165-218.
6. Birgin, E., Martínez, J. & Raydan, M. (2000) Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim., 10, 1196-1211.
7. Biswas, P., Lian, T. C., Wang, T. C. & Ye, Y. (2006) Semidefinite programming based algorithms for sensor network localization. ACM Trans. Sen. Netw., 2, 188-220.
8. Borg, I. & Groenen, P. (2010) Modern Multidimensional Scaling. New York, NY: Springer Science.
9. Boufounos, P. & Baraniuk, R. (2008) 1-Bit compressive sensing. Proceedings of IEEE Conference on Informational Science and Systems (CISS), Princeton, NJ, USA.
10. Boucheron, S., Bousquet, O. & Lugosi, G. (2005) Theory of classification: a survey of some recent advances. ESAIM: Probab. Stat., 9, 323-375.
11. 2 penalization. Electron. J. Stat., 2, 1153-1194.
12. Cai, J.-F., Candès, E. & Shen, Z. (2010) A singular value thresholding algorithm for matrix completion. SIAM J. Optim., 20, 1956-1982.
13. Candès, E. (2006) Compressive sampling. Proceedings of International Congress of Mathematics, Madrid, Spain.
14. Candès, E. & Davenport, M. (2013) How well can we estimate a sparse vector? Appl. Comput. Harmon. Anal., 34, 317-323.
15. Candès, E. & Plan, Y. (2010) Matrix completion with noise. Proc. IEEE, 98, 925-936.
16. Candès, E. & Recht, B. (2009) Exact matrix completion via convex optimization. Found. Comput. Math., 9, 717-772.
17. Candès, E. & Tao, T. (2010) The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inform. Theory, 56, 2053-2080.
18. Chatterjee, S. (2012) Matrix estimation by universal singular value thresholding. Arxiv preprint 1212. 1247.
19. Collins, M., Dasgupta, S. & Schapire, R. (2001) A generalization of principal component analysis to the exponential family. Proceedings of Advances in Neural Processing Systems (NIPS), Vancouver, BC, Canada.
20. Cover, T. & Thomas, J. Elements of Information Theory. New York, NY: Wiley-Interscience, 1991.
21. Davenport, M., Duarte, M., Eldar, Y. & Kutyniok, G. (2012) Introduction to compressed sensing. Compressed Sensing: Theory and Applications (Y. Eldar & G. Kutyniok eds). Cambridge, UK: Cambridge University Press.
22. De Leeuw, J. (2006) Principal component analysis of binary data by iterated singular value decomposition. Comput. Stat. Data Anal., 50, 21-39.
23. Donoho, D. (2006) Compressed sensing. IEEE Trans. Inform. Theory, 52, 1289-1306.
24. Gabay, D. & Mercier, B. (1976) A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl., 2, 17-40.
25. Gaïffas, S. & Lecué, G. (2010) Sharp oracle inequalities for the prediction of a high-dimensional matrix. Arxiv preprint 1008.4886.
26. Gleich, D. & Lim, L.-H. (2011) Rank aggregation via nuclear norm minimization. Proceedings of ACM SIGKDD International Conference on Knowledge, Discovery, and Data Mining (KDD), San Diego, CA, USA.
27. Glowinski, R. & Marrocco, A. (1975) Sur l'approximation, par elements finis d'ordre un, et la resolution, par penalisation-dualité, d'une classe de problems de Dirichlet non lineares. Revue Française d'Automatique, Informatique, et Recherche Opérationelle, 9, 41-76.
28. Goldberg, D., Nichols, D., Oki, B. & Terry, D. (1992) Using collaborative filtering to weave an information tapestry. Comm. ACM, 35, 61-70.
29. Green, P. & Wind, Y. (1973) Multivariate Decisions in Marketing: A Measurement Approach. Hinsdale, IL: Dryden.
30. Gross, D. (2011) Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theory, 57, 1548-1566.
31. Gross, D., Liu, Y., Flammia, S., Becker, S. & Eisert, J. (2010) Quantum state tomography via compressed sensing. Phys. Rev. Lett., 105, 150401.
32. Gupta, A., Nowak, R. & Recht, B. (2010) Sample complexity for 1-bit compressed sensing and sparse classification. Proc. IEEE Int. Symp. Inform. Theory (ISIT), Austin, TX, USA.
33. Jacques, L., Laska, J., Boufounos, P. & Baraniuk, R. (2013) Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors. IEEE Trans. Inform. Theory, 59, 2082-2102.
34. Kakade, S., Shamir, O., Sridharan, K. & Tewari, A. (2009) Learning exponential families in high-dimensions: strong convexity and sparsity. Proceedings of International Conference on Art. Intell. Stat. (AISTATS), Clearwater Beach, FL, USA.
35. Kar, P., Sriperumbudur, B., Jain, P. & Karnick, H. (2013) On the generalization ability of online learning algorithms for pairwise loss functions. Proceedings of International Conference on Machine Learning (ICML), Atlanta, GA, USA.
36. Keshavan, R., Montanari, A. & Oh, S. (2010) Matrix completion from a few entries. IEEE Trans. Inform. Theory, 56, 2980-2998.
37. Keshavan, R., Montanari, A. & Oh, S. (2010) Matrix completion from noisy entries. J. Machine Learning Research, 11, 2057-2078.
38. Khan, M., Marlin, B., Bouchard, G. & Murphy, K. (2010) Variational bounds for mixed-data factor analysis. Proceedings of Advances in Neural Processing Systems (NIPS), Vancouver, BC, Canada.
39. Klopp, O. (2011) High dimensional matrix estimation with unknown variance of the noise. Arxiv preprint 1112.3055.
40. Klopp, O. (2011) Rank penalized estimators for high-dimensional matrices. Electron. J. Stat., 5, 1161-1183.
41. Koltchinskii, V. (2012) Von Neumann entropy penalization and low-rank matrix estimation. Ann. Stat., 39, 2936-2973.
42. Koltchinskii, V., Lounici, K. & Tsybakov, A. (2011) Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Stat., 39, 2302-2329.
43. Laska, J. & Baraniuk, R. (2012) Regime change: bit-depth versus measurement-rate in compressive sensing. IEEE Trans. Signal Process., 60, 3496-3505.
44. Ledoux, M. & Talagrand, M. (1991) Probability in Banach Spaces: Isoperimetry and Processes. Berlin, Germany: Springer.
45. Liu, Z. & Vandenberghe, L. (2009) Interior-point method for nuclear norm approximation with application to system identification. SIAM J. Matrix Anal. Appl., 31, 1235-1256.
46. Meier, L., Van De Geer, S. & Bühlmann, P. (2008) The group lasso for logistic regression. J. R. Stat. Soc. B, 70, 53-71.
47. Miller, G. (1956) The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychol. Rev., 63, 81-97.
48. Negahban, S., Ravikumar, P., Wainwright, M. & Yu, B. (2012) A unified framework for high-dimensional analysis of m-estimators with decomposable regularizers. Stat. Sci., 27, 538-557.
49. Negahban, S. & Wainwright, M. (2012) Restricted strong convexity and weighted matrix completion: optimal bounds with noise. J. Mach. Learn. Res., 13, 1665-1697.
50. Nesterov, Y. (2004) Introductory Lectures on Convex Optimization. Dordrecht, The Netherlands: Kluwer Academic Publisher.
51. Plan, Y. & Vershynin, R. (2013) One-bit compressed sensing by linear programming. Comm. Pure Appl. Math. 66, 1275-1297.
52. Plan, Y. & Vershynin, R. (2013) Robust 1-bit compressed sensing and sparse logistic regression: a convex programming approach. Proc. IEEE Trans. Inform. Theory, 59, 482-494.
53. q-balls. IEEE Trans. Inform. Theory, 57, 6976-6994.
54. 1-regularized logistic regression. Ann. Stat., 38, 1287-1319.
55. Recht, B. (2011) A simpler approach to matrix completion. J. Mach. Learn. Res., 12, 3413-3430.
56. Rohde, A. & Tsybakov, A. (2011) Estimation of high-dimensional low-rank matrices. Ann. Stat., 39, 887-930.
57. Schein, A., Saul, L. & Ungar, L. (2003) A generalized linear model for principal component analysis of binary data. Proc. Int. Conf. Art. Intell. Stat. (AISTATS), Key West, FL, USA.
58. Seginer, Y. (2000) The expected norm of random matrices. Comb. Probab. Comput., 9, 149-166.
59. Singer, A. (2008) A remark on global positioning from local distances. Proc. Natl. Acad. Sci., 105, 9507-9511.
60. Singer, A. & Cucuringu, M. (2010) Uniqueness of low-rank matrix completion by rigidity theory. SIAM J. Matrix Anal. Appl., 31, 1621-1641.
61. Spence, I. & Domoney, D. (1974) Single subject incomplete designs for nonmetric multidimensional scaling. Psychometrika, 39, 469-490.
62. Srebro, N., Alon, N. & Jaakkola, T. (2004) Generalization error bounds for collaborative prediction with low-rank matrices. Proceedings of Advances in Neural Processing Systems (NIPS), Vancouver, BC, Canada.
63. Srebro, N., Rennie, J. D. & Jaakkola, T. (2004) Maximum-margin matrix factorization. Proceedings of Advances in Neural Processing Systems (NIPS), Vancouver, BC, Canada.
64. Tipping, M. (1998) Probabilistic visualization of high-dimensional binary data. Proceedings of Advances in Neural Processing Systems (NIPS), Vancouver, BC, Canada.
65. Van De Geer, S. (2008) High-dimensional generalized linear models and the lasso. Ann. Stat., 36, 614-645.
66. van den Berg, E. & Friedlander, M. P. (2007) SPGL1. http://www.cs.ubc.ca/∼mpf/spgl1/.
67. van den Berg, E. & Friedlander, M. P. (2008) Probing the Pareto frontier for basis-pursuit solutions. SIAM J. Sci. Comput., 31, 890-912.
68. Zymnis, A., Boyd, S. & Candès, E. (2010) Compressed sensing with quantized measurements. IEEE Signal Process. Lett., 17, 149-152.