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Volumn 12, Issue 4, 2012, Pages 389-434

User-Friendly Tail Bounds for Sums of Random Matrices

Author keywords

Discrete time martingale; Large deviation; Probability inequality; Random matrix; Sum of independent random variables

Indexed keywords


EID: 84864315555     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-011-9099-z     Document Type: Article
Times cited : (1499)

References (58)
  • 1
    • 67650105330 scopus 로고    scopus 로고
    • The fast Johnson-Lindenstrauss transform and approximate nearest neighbors
    • N. Ailon, B. Chazelle, The fast Johnson-Lindenstrauss transform and approximate nearest neighbors, SIAM J. Comput. 39(1), 302-322 (2009).
    • (2009) SIAM J. Comput. , vol.39 , Issue.1 , pp. 302-322
    • Ailon, N.1    Chazelle, B.2
  • 2
    • 34247228045 scopus 로고    scopus 로고
    • Fast computation of low-rank matrix approximations
    • Article 10(electronic)
    • D. Achlioptas, F. McSherry, Fast computation of low-rank matrix approximations, J. Assoc. Comput. Mach. 54(2), Article 10 (2007) (electronic).
    • (2007) J. Assoc. Comput. Mach. , vol.54 , Issue.2
    • Achlioptas, D.1    McSherry, F.2
  • 3
    • 0036495139 scopus 로고    scopus 로고
    • Strong converse for identification via quantum channels
    • R. Ahlswede, A. Winter, Strong converse for identification via quantum channels, IEEE Trans. Inf. Theory48(3), 569-579 (2002).
    • (2002) IEEE Trans. Inf. Theory , vol.48 , Issue.3 , pp. 569-579
    • Ahlswede, R.1    Winter, A.2
  • 4
    • 84864293092 scopus 로고    scopus 로고
    • Graduate Texts in Mathematics, Berlin: Springer
    • R. Bhatia, Matrix Analysis. Graduate Texts in Mathematics, vol. 169 (Springer, Berlin, 1997), p. 10.
    • (1997) Matrix Analysis , vol.169 , pp. 10
    • Bhatia, R.1
  • 6
    • 0004124953 scopus 로고    scopus 로고
    • Providence: American Mathematical Society
    • V. Bogdanov, Gaussian Measures (American Mathematical Society, Providence, 1998).
    • (1998) Gaussian Measures
    • Bogdanov, V.1
  • 7
    • 0035607466 scopus 로고    scopus 로고
    • Operator Khintchine inequality in non-commutative probability
    • A. Buchholz, Operator Khintchine inequality in non-commutative probability, Math. Ann. 319, 1-16 (2001).
    • (2001) Math. Ann. , vol.319 , pp. 1-16
    • Buchholz, A.1
  • 8
    • 42149120579 scopus 로고    scopus 로고
    • Optimal constants in Khintchine-type inequalities for Fermions, Rademachers and q-Gaussian operators
    • A. Buchholz, Optimal constants in Khintchine-type inequalities for Fermions, Rademachers and q-Gaussian operators, Bull. Pol. Acad. Sci., Math. 53(3), 315-321 (2005).
    • (2005) Bull. Pol. Acad. Sci., Math. , vol.53 , Issue.3 , pp. 315-321
    • Buchholz, A.1
  • 9
    • 0000182415 scopus 로고
    • A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations
    • H. Chernoff, A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Stat. 23(4), 493-507 (1952).
    • (1952) Ann. Math. Stat. , vol.23 , Issue.4 , pp. 493-507
    • Chernoff, H.1
  • 10
    • 38049175146 scopus 로고    scopus 로고
    • Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales
    • D. Cristofides, K. Markström, Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales, Random Struct. Algorithms32(8), 88-100 (2008).
    • (2008) Random Struct. Algorithms , vol.32 , Issue.8 , pp. 88-100
    • Cristofides, D.1    Markström, K.2
  • 11
    • 34249687049 scopus 로고    scopus 로고
    • Sparsity and incoherence in compressive sampling
    • E. Candès, J. K. Romberg, Sparsity and incoherence in compressive sampling, Inverse Probl. 23(3), 969-985 (2007).
    • (2007) Inverse Probl. , vol.23 , Issue.3 , pp. 969-985
    • Candès, E.1    Romberg, J.K.2
  • 13
    • 0012070851 scopus 로고    scopus 로고
    • Local operator theory, random matrices, and Banach spaces
    • W. B. Johnson and J. Lindenstrauss (Eds.), Amsterdam: Elsevier
    • K. R. Davidson, S. J. Szarek. Local operator theory, random matrices, and Banach spaces, in Handbook of Banach Space Geometry, ed. by W. B. Johnson, J. Lindenstrauss (Elsevier, Amsterdam, 2002), pp. 317-366.
    • (2002) Handbook of Banach Space Geometry , pp. 317-366
    • Davidson, K.R.1    Szarek, S.J.2
  • 15
    • 59049088495 scopus 로고    scopus 로고
    • A matrix convexity approach to some celebrated quantum inequalities
    • E. G. Effros, A matrix convexity approach to some celebrated quantum inequalities, Proc. Natl. Acad. Sci. USA106(4), 1006-1008 (2009).
    • (2009) Proc. Natl. Acad. Sci. USA , vol.106 , Issue.4 , pp. 1006-1008
    • Effros, E.G.1
  • 16
    • 0000230165 scopus 로고
    • Remarks on two theorems of E. Lieb
    • H. Epstein, Remarks on two theorems of E. Lieb, Commun. Math. Phys. 31, 317-325 (1973).
    • (1973) Commun. Math. Phys. , vol.31 , pp. 317-325
    • Epstein, H.1
  • 17
    • 0002384441 scopus 로고
    • On tail probabilities for martingales
    • D. A. Freedman, On tail probabilities for martingales, Ann. Probab. 3(1), 100-118 (1975).
    • (1975) Ann. Probab. , vol.3 , Issue.1 , pp. 100-118
    • Freedman, D.A.1
  • 18
    • 51249179500 scopus 로고
    • Some inequalities for Gaussian processes and applications
    • Y. Gordon, Some inequalities for Gaussian processes and applications, Isr. J. Math. 50(4), 265-289 (1985).
    • (1985) Isr. J. Math. , vol.50 , Issue.4 , pp. 265-289
    • Gordon, Y.1
  • 19
    • 0011597392 scopus 로고
    • Majorization of Gaussian processes and geometric applications
    • Y. Gordon, Majorization of Gaussian processes and geometric applications, Probab. Theory Relat. Fields91(2), 251-267 (1992).
    • (1992) Probab. Theory Relat. Fields , vol.91 , Issue.2 , pp. 251-267
    • Gordon, Y.1
  • 20
    • 79951886985 scopus 로고    scopus 로고
    • Recovering low-rank matrices from few coefficients in any basis
    • D. Gross, Recovering low-rank matrices from few coefficients in any basis, IEEE Trans. Inf. Theory57(3), 1548-1566 (2011).
    • (2011) IEEE Trans. Inf. Theory , vol.57 , Issue.3 , pp. 1548-1566
    • Gross, D.1
  • 24
    • 79960425522 scopus 로고    scopus 로고
    • Finding structure with randomness: stochastic algorithms for constructing approximate matrix decompositions
    • N. Halko, P.-G. Martinsson, J. A. Tropp, Finding structure with randomness: stochastic algorithms for constructing approximate matrix decompositions, SIAM Rev. 53(2), 217-288 (2011).
    • (2011) SIAM Rev. , vol.53 , Issue.2 , pp. 217-288
    • Halko, N.1    Martinsson, P.-G.2    Tropp, J.A.3
  • 26
    • 29144436414 scopus 로고    scopus 로고
    • On the best constants in some non-commutative martingale inequalities
    • M. Junge, Q. Xu, On the best constants in some non-commutative martingale inequalities, Bull. Lond. Math. Soc. 37, 243-253 (2005).
    • (2005) Bull. Lond. Math. Soc. , vol.37 , pp. 243-253
    • Junge, M.1    Xu, Q.2
  • 27
    • 57849107738 scopus 로고    scopus 로고
    • Noncommutative Burkholder/Rosenthal inequalities II: Applications
    • M. Junge, Q. Xu, Noncommutative Burkholder/Rosenthal inequalities II: Applications, Isr. J. Math. 167, 227-282 (2008).
    • (2008) Isr. J. Math. , vol.167 , pp. 227-282
    • Junge, M.1    Xu, Q.2
  • 28
    • 18144366601 scopus 로고    scopus 로고
    • Some estimates of norms of random matrices
    • R. Latała, Some estimates of norms of random matrices, Proc. Am. Math. Soc. 133(5), 1273-1282 (2005).
    • (2005) Proc. Am. Math. Soc. , vol.133 , Issue.5 , pp. 1273-1282
    • Latała, R.1
  • 29
    • 36649022319 scopus 로고
    • Convex trace functions and the Wigner-Yanase-Dyson conjecture
    • E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. Math. 11, 267-288 (1973).
    • (1973) Adv. Math. , vol.11 , pp. 267-288
    • Lieb, E.H.1
  • 30
    • 0001335497 scopus 로고
    • Expectations and entropy inequalities for finite quantum systems
    • G. Lindblad, Expectations and entropy inequalities for finite quantum systems, Commun. Math. Phys. 39, 111-119 (1974).
    • (1974) Commun. Math. Phys. , vol.39 , pp. 111-119
    • Lindblad, G.1
  • 32
    • 0000426206 scopus 로고
    • Noncommutative Khintchine and Paley inequalities
    • F. Lust-Piquard, G. Pisier, Noncommutative Khintchine and Paley inequalities, Ark. Mat. 29(2), 241-260 (1991).
    • (1991) Ark. Mat. , vol.29 , Issue.2 , pp. 241-260
    • Lust-Piquard, F.1    Pisier, G.2
  • 38
    • 33846707343 scopus 로고    scopus 로고
    • Sums of random symmetric matrices and quadratic optimization under orthogonality constraints
    • A. Nemirovski, Sums of random symmetric matrices and quadratic optimization under orthogonality constraints, Math. Program., Ser. B109, 283-317 (2007).
    • (2007) Math. Program., Ser. B , vol.109 , pp. 283-317
    • Nemirovski, A.1
  • 40
    • 77955434395 scopus 로고    scopus 로고
    • Sums of random Hermitian matrices and an inequality by Rudelson
    • R. I. Oliveira, Sums of random Hermitian matrices and an inequality by Rudelson, Electron. Commun. Probab. 15, 203-212 (2010).
    • (2010) Electron. Commun. Probab. , vol.15 , pp. 203-212
    • Oliveira, R.I.1
  • 41
    • 0003668665 scopus 로고
    • Classics in Applied Mathematics, Philadelphia: Society for Industrial and Applied Mathematics
    • B. N. Parlett, The Symmetric Eigenvalue Problem. Classics in Applied Mathematics, vol. 20 (Society for Industrial and Applied Mathematics, Philadelphia, 1987).
    • (1987) The Symmetric Eigenvalue Problem , vol.20
    • Parlett, B.N.1
  • 42
    • 0242554667 scopus 로고    scopus 로고
    • Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge Univ. Press
    • V. I. Paulsen, Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics, vol. 78 (Cambridge Univ. Press, Cambridge, 2002).
    • (2002) Completely Bounded Maps and Operator Algebras , vol.78
    • Paulsen, V.I.1
  • 43
    • 0012475858 scopus 로고
    • A survey of certain trace inequalities
    • Banach Center Publications, Warsaw: Polish Acad. Sci
    • D. Petz, A survey of certain trace inequalities, in Functional Analysis and Operator Theory. Banach Center Publications, vol. 30 (Polish Acad. Sci., Warsaw, 1994), pp. 287-298.
    • (1994) Functional Analysis and Operator Theory , vol.30 , pp. 287-298
    • Petz, D.1
  • 45
    • 79551670254 scopus 로고    scopus 로고
    • Simpler approach to matrix completion
    • to appear. Available at
    • B. Recht, Simpler approach to matrix completion, J. Mach. Learn. Res. (2009, to appear). Available at http://pages. cs. wisc. edu/brecht/papers/09. Recht. ImprovedMC. pdf.
    • (2009) J. Mach. Learn. Res.
    • Recht, B.1
  • 46
    • 0033541884 scopus 로고    scopus 로고
    • Random vectors in the isotropic position
    • M. Rudelson, Random vectors in the isotropic position, J. Funct. Anal. 164, 60-72 (1999).
    • (1999) J. Funct. Anal. , vol.164 , pp. 60-72
    • Rudelson, M.1
  • 47
    • 0035981724 scopus 로고    scopus 로고
    • Inequalities for quantum entropy: a review with conditions for equality
    • Erratum: J. Math. Phys.46(1), 0199101 (2005)
    • M. B. Ruskai, Inequalities for quantum entropy: a review with conditions for equality, J. Math. Phys. 43(9), 4358-4375 (2002). Erratum: J. Math. Phys. 46(1), 0199101 (2005).
    • (2002) J. Math. Phys. , vol.43 , Issue.9 , pp. 4358-4375
    • Ruskai, M.B.1
  • 48
    • 34547728320 scopus 로고    scopus 로고
    • Sampling from large matrices: an approach through geometric functional analysis
    • Article 21(electronic) 19 pp
    • M. Rudelson, R. Vershynin, Sampling from large matrices: an approach through geometric functional analysis, J. Assoc. Comput. Mach. 54(4), Article 21 (2007) (electronic) 19 pp.
    • (2007) J. Assoc. Comput. Mach. , vol.54 , Issue.4
    • Rudelson, M.1    Vershynin, R.2
  • 49
    • 0034382620 scopus 로고    scopus 로고
    • The expected norm of random matrices
    • Y. Seginer, The expected norm of random matrices, Comb. Probab. Comput. 9, 149-166 (2000).
    • (2000) Comb. Probab. Comput. , vol.9 , pp. 149-166
    • Seginer, Y.1
  • 50
    • 84864318742 scopus 로고    scopus 로고
    • Moment inequalities for sums of random matrices and their applications in optimization
    • (electronic)
    • A. M.-C. So, Moment inequalities for sums of random matrices and their applications in optimization, Math. Prog. Ser. A (2009) (electronic).
    • (2009) Math. Prog. Ser. A
    • So, A.M.-C.1
  • 51
    • 34247360062 scopus 로고    scopus 로고
    • Smoothed analysis of the condition numbers and growth factors of matrices
    • A. Sankar, D. A. Spielman, S.-H. Teng, Smoothed analysis of the condition numbers and growth factors of matrices, SIAM J. Matrix Anal. Appl. 28(2), 446-476 (2006).
    • (2006) SIAM J. Matrix Anal. Appl. , vol.28 , Issue.2 , pp. 446-476
    • Sankar, A.1    Spielman, D.A.2    Teng, S.-H.3
  • 53
    • 44249104733 scopus 로고    scopus 로고
    • On the conditioning of random subdictionaries
    • J. A. Tropp, On the conditioning of random subdictionaries, Appl. Comput. Harmon. Anal. 25, 1-24 (2008).
    • (2008) Appl. Comput. Harmon. Anal. , vol.25 , pp. 1-24
    • Tropp, J.A.1
  • 54
    • 80052645998 scopus 로고    scopus 로고
    • Improved analysis of the subsampled randomized Hadamard transform
    • to appear. Available at arXiv: 1011. 1595
    • J. A. Tropp, Improved analysis of the subsampled randomized Hadamard transform. Adv. Adapt. Data Anal. (2010, to appear). Available at arXiv: 1011. 1595.
    • (2010) Adv. Adapt. Data Anal.
    • Tropp, J.A.1
  • 55
    • 84859402279 scopus 로고    scopus 로고
    • Freedman's inequality for matrix martingales
    • J. A. Tropp, Freedman's inequality for matrix martingales, Electron. Commun. Probab. 16, 262-270 (2011).
    • (2011) Electron. Commun. Probab. , vol.16 , pp. 262-270
    • Tropp, J.A.1
  • 56
    • 84864302110 scopus 로고    scopus 로고
    • From the joint convexity of quantum relative entropy to a concavity theorem of Lieb
    • to appear). Available at arXiv: 1101. 1070
    • J. A. Tropp, From the joint convexity of quantum relative entropy to a concavity theorem of Lieb. Proc. Amer. Math. Soc. (2011, to appear). Available at arXiv: 1101. 1070.
    • (2011) Proc. Amer. Math. Soc.
    • Tropp, J.A.1


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