Deterministic algorithms for matrix completion
ABSTRACT The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application areas, and has received a great deal of attention in the context of the Netflix challenge. This setup usually represents our partial...
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| Vydané v: | Random structures & algorithms Ročník 45; číslo 2; s. 306 - 317 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Hoboken
Blackwell Publishing Ltd
01.09.2014
Wiley Subscription Services, Inc |
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| ISSN: | 1042-9832, 1098-2418 |
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| Abstract | ABSTRACT
The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application areas, and has received a great deal of attention in the context of the Netflix challenge. This setup usually represents our partial knowledge of some information domain. Unknown entries may be due to the unavailability of some relevant experimental data.
One approach to this problem starts by selecting a complexity measure of matrices, such as rank or trace norm. The corresponding algorithm outputs a matrix of lowest possible complexity that agrees with the partially specified matrix. The performance of the above algorithm under the assumption that the revealed entries are sampled randomly has received considerable attention (e.g., Refs. Srebro et al., 2005; COLT, 2005; Foygel and Srebro, 2011; Candes and Tao, 2010; Recht, 2009; Keshavan et al., 2010; Koltchinskii et al., 2010). Here we ask what can be said if the observed entries are chosen deterministically. We prove generalization error bounds for such deterministic algorithms, that resemble the results of Refs. Srebro et al. (2005); COLT (2005); Foygel and Srebro (2011) for the randomized algorithms.
We still do not understand which sets of entries in a given matrix can be used to properly reconstruct it. Our hope is that the present work sheds some light on this problem. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 306–317, 2014 |
|---|---|
| AbstractList | The goal of the
matrix completion problem
is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application areas, and has received a great deal of attention in the context of the
Netflix challenge
. This setup usually represents our partial knowledge of some information domain. Unknown entries may be due to the unavailability of some relevant experimental data.
One approach to this problem starts by selecting a complexity measure of matrices, such as rank or trace norm. The corresponding algorithm outputs a matrix of lowest possible complexity that agrees with the partially specified matrix. The performance of the above algorithm under the assumption that the revealed entries are sampled randomly has received considerable attention (e.g., Refs. Srebro et al., 2005; COLT, 2005; Foygel and Srebro, 2011; Candes and Tao, 2010; Recht, 2009; Keshavan et al., 2010; Koltchinskii et al., 2010). Here we ask what can be said if the observed entries are chosen deterministically. We prove generalization error bounds for such deterministic algorithms, that resemble the results of Refs. Srebro et al. (2005); COLT (2005); Foygel and Srebro (2011) for the randomized algorithms.
We still do not understand which sets of entries in a given matrix can be used to properly reconstruct it. Our hope is that the present work sheds some light on this problem. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 306–317, 2014 The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application areas, and has received a great deal of attention in the context of the Netflix challenge. This setup usually represents our partial knowledge of some information domain. Unknown entries may be due to the unavailability of some relevant experimental data. One approach to this problem starts by selecting a complexity measure of matrices, such as rank or trace norm. The corresponding algorithm outputs a matrix of lowest possible complexity that agrees with the partially specified matrix. The performance of the above algorithm under the assumption that the revealed entries are sampled randomly has received considerable attention (e.g., Refs. Srebro et al., 2005; COLT, 2005; Foygel and Srebro, 2011; Candes and Tao, 2010; Recht, 2009; Keshavan et al., 2010; Koltchinskii et al., 2010). Here we ask what can be said if the observed entries are chosen deterministically. We prove generalization error bounds for such deterministic algorithms, that resemble the results of Refs. Srebro et al. (2005); COLT (2005); Foygel and Srebro (2011) for the randomized algorithms. We still do not understand which sets of entries in a given matrix can be used to properly reconstruct it. Our hope is that the present work sheds some light on this problem. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 306-317, 2014 [PUBLICATION ABSTRACT] The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application areas, and has received a great deal of attention in the context of the Netflix challenge. This setup usually represents our partial knowledge of some information domain. Unknown entries may be due to the unavailability of some relevant experimental data. One approach to this problem starts by selecting a complexity measure of matrices, such as rank or trace norm. The corresponding algorithm outputs a matrix of lowest possible complexity that agrees with the partially specified matrix. The performance of the above algorithm under the assumption that the revealed entries are sampled randomly has received considerable attention (e.g., Refs. Srebro et al., 2005; COLT, 2005; Foygel and Srebro, 2011; Candes and Tao, 2010; Recht, 2009; Keshavan et al., 2010; Koltchinskii et al., 2010). Here we ask what can be said if the observed entries are chosen deterministically. We prove generalization error bounds for such deterministic algorithms, that resemble the results of Refs. Srebro et al. (2005); COLT (2005); Foygel and Srebro (2011) for the randomized algorithms. We still do not understand which sets of entries in a given matrix can be used to properly reconstruct it. Our hope is that the present work sheds some light on this problem. copyright 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 306-317, 2014 ABSTRACT The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application areas, and has received a great deal of attention in the context of the Netflix challenge. This setup usually represents our partial knowledge of some information domain. Unknown entries may be due to the unavailability of some relevant experimental data. One approach to this problem starts by selecting a complexity measure of matrices, such as rank or trace norm. The corresponding algorithm outputs a matrix of lowest possible complexity that agrees with the partially specified matrix. The performance of the above algorithm under the assumption that the revealed entries are sampled randomly has received considerable attention (e.g., Refs. Srebro et al., 2005; COLT, 2005; Foygel and Srebro, 2011; Candes and Tao, 2010; Recht, 2009; Keshavan et al., 2010; Koltchinskii et al., 2010). Here we ask what can be said if the observed entries are chosen deterministically. We prove generalization error bounds for such deterministic algorithms, that resemble the results of Refs. Srebro et al. (2005); COLT (2005); Foygel and Srebro (2011) for the randomized algorithms. We still do not understand which sets of entries in a given matrix can be used to properly reconstruct it. Our hope is that the present work sheds some light on this problem. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 306–317, 2014 |
| Author | Shraibman, Adi Heiman, Eyal Schechtman, Gideon |
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| References | V. Koltchinskii, A. B. Tsybakov, and K. Lounici, Nuclear norm penalization and optimal rates for noisy low rank matrix completion, Ann Statist 39 (2011), 2302-2329. E.J. Candes and B. Recht, Exact matrix completion via convex optimization, Found Comput Math 9 (2009), 717-772. S. Hoory, N. Linial, and A. Wigderson, Expander graphs and their applications, Bull Am Math Soc 43 (2006), 439-562. G. J. O. Jameson, Summing and nuclear norms in banach space theory, Cambridge University Press, 1987. R. H. Keshavan, A. Montanari, and S. Oh, Matrix completion from noisy entries, J Mach Learn Res 11 (2010), 2057-2078. A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261-277. B. Recht, A simpler approach to matrix completion, J Machine Learn Res, 12 (2011), 3413-3430. E. J. Candes and T. Tao, The power of convex relaxation: near-optimal matrix completion, IEEE Trans Infor Theo 56 (2010), 2053-2080. 2010; 56 2010; 11 2001; 6 2011 2006; 43 1988; 8 2009 1987 2008 1986 1996 2009; 9 2005 2011; 12 2011; 39 2005; 17 1989 Pisier G. (e_1_2_6_15_1) 1986 Foygel R. (e_1_2_6_7_1) 2011 Tomczak‐Jaegermann N. (e_1_2_6_19_1) 1989 Keshavan R. H. (e_1_2_6_10_1) 2010; 11 Recht B. (e_1_2_6_16_1) 2011; 12 Fazel M. (e_1_2_6_6_1) 2001 e_1_2_6_9_1 Lee T. (e_1_2_6_13_1) 2008 e_1_2_6_8_1 Srebro N. (e_1_2_6_18_1) 2005 Benczúr A.A. (e_1_2_6_3_1) 1996 Srebro N. (e_1_2_6_17_1) 2005 e_1_2_6_5_1 Lee T. (e_1_2_6_12_1) 2008 e_1_2_6_4_1 e_1_2_6_14_1 e_1_2_6_11_1 e_1_2_6_2_1 |
| References_xml | – reference: B. Recht, A simpler approach to matrix completion, J Machine Learn Res, 12 (2011), 3413-3430. – reference: S. Hoory, N. Linial, and A. Wigderson, Expander graphs and their applications, Bull Am Math Soc 43 (2006), 439-562. – reference: E. J. Candes and T. Tao, The power of convex relaxation: near-optimal matrix completion, IEEE Trans Infor Theo 56 (2010), 2053-2080. – reference: E.J. Candes and B. Recht, Exact matrix completion via convex optimization, Found Comput Math 9 (2009), 717-772. – reference: G. J. O. Jameson, Summing and nuclear norms in banach space theory, Cambridge University Press, 1987. – reference: A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261-277. – reference: R. H. Keshavan, A. Montanari, and S. Oh, Matrix completion from noisy entries, J Mach Learn Res 11 (2010), 2057-2078. – reference: V. Koltchinskii, A. B. Tsybakov, and K. Lounici, Nuclear norm penalization and optimal rates for noisy low rank matrix completion, Ann Statist 39 (2011), 2302-2329. – year: 2011 – year: 1986 – start-page: 255 year: 2009 end-page: 262 – volume: 11 start-page: 2057 year: 2010 end-page: 2078 article-title: Matrix completion from noisy entries publication-title: J Mach Learn Res – volume: 56 start-page: 2053 year: 2010 end-page: 2080 article-title: The power of convex relaxation: near‐optimal matrix completion publication-title: IEEE Trans Infor Theo – start-page: 71 year: 2008 end-page: 80 – year: 1987 – volume: 39 start-page: 2302 year: 2011 end-page: 2329 article-title: Nuclear norm penalization and optimal rates for noisy low rank matrix completion publication-title: Ann Statist – volume: 17 start-page: 1329 year: 2005 end-page: 1336 – year: 1989 – volume: 9 start-page: 717 year: 2009 end-page: 772 article-title: Exact matrix completion via convex optimization publication-title: Found Comput Math – volume: 43 start-page: 439 year: 2006 end-page: 562 article-title: Expander graphs and their applications publication-title: Bull Am Math Soc – volume: 8 start-page: 261 year: 1988 end-page: 277 article-title: Ramanujan graphs publication-title: Combinatorica – start-page: 47 year: 1996 end-page: 55 – volume: 12 start-page: 3413 year: 2011 end-page: 3430 article-title: A simpler approach to matrix completion publication-title: J Machine Learn Res – start-page: 545 year: 2005 end-page: 560 – start-page: 351 year: 2008 end-page: 357 – volume: 6 start-page: 4734 year: 2001 end-page: 4739 – ident: e_1_2_6_2_1 doi: 10.1145/1536414.1536451 – start-page: 71 volume-title: Proceedings of the 23rd IEEE Conference on Computational Complexity year: 2008 ident: e_1_2_6_13_1 – start-page: 1329 volume-title: Advances in Neural Information Processing Systems 17 year: 2005 ident: e_1_2_6_17_1 – volume: 11 start-page: 2057 year: 2010 ident: e_1_2_6_10_1 article-title: Matrix completion from noisy entries publication-title: J Mach Learn Res – ident: e_1_2_6_11_1 doi: 10.1214/11-AOS894 – start-page: 351 volume-title: Proceedings of the 24th IEEE Conference on Computational Complexity year: 2008 ident: e_1_2_6_12_1 – ident: e_1_2_6_5_1 doi: 10.1109/TIT.2010.2044061 – ident: e_1_2_6_8_1 doi: 10.1090/S0273-0979-06-01126-8 – volume-title: Vol. 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics year: 1989 ident: e_1_2_6_19_1 – volume-title: 24th Annual Conference on Learning Theory (COLT) year: 2011 ident: e_1_2_6_7_1 – ident: e_1_2_6_9_1 doi: 10.1017/CBO9780511569166 – ident: e_1_2_6_4_1 doi: 10.1007/s10208-009-9045-5 – volume: 12 start-page: 3413 year: 2011 ident: e_1_2_6_16_1 article-title: A simpler approach to matrix completion publication-title: J Machine Learn Res – volume-title: Published for the Conference Board of the Mathematical Sciences year: 1986 ident: e_1_2_6_15_1 – start-page: 4734 volume-title: Proceedings American Control Conference year: 2001 ident: e_1_2_6_6_1 – start-page: 545 volume-title: 18th Annual Conference on Computational Learning Theory (COLT) year: 2005 ident: e_1_2_6_18_1 – start-page: 47 volume-title: STOC '1996: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (May 1996) year: 1996 ident: e_1_2_6_3_1 – ident: e_1_2_6_14_1 doi: 10.1007/BF02126799 |
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The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many... The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application... The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application... |
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| SubjectTerms | Algorithms Complexity deterministic guarantees Errors expander graphs factorization norms graph sparsifiers matrix completion Norms Sheds |
| Title | Deterministic algorithms for matrix completion |
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