Sums of random Hermitian matrices and an inequality by Rudelson

We give a new, elementary proof of a key inequality used by Rudelson in the derivation of his well-known bound for random sums of rank-one operators. Our approach is based on Ahlswede and Winter's technique for proving operator Chernoff bounds. We also prove a concentration inequality for sums of random matrices of rank one with explicit constants.

[1]  S. Golden.  LOWER BOUNDS FOR THE HELMHOLTZ FUNCTION , 1965 .

[2]  C. Thompson.  Inequality with Applications in Statistical Mechanics , 1965 .

[3]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[4]  M. Talagrand,et al.  Probability in Banach spaces , 1991 .

[5]  G. Pisier,et al.  Non commutative Khintchine and Paley inequalities , 1991 .

[6]  M. Rudelson.  Random Vectors in the Isotropic Position , 1996, math/9608208.

[7]  A. Buchholz.  Operator Khintchine inequality in non-commutative probability , 2001 .

[8]  Rudolf Ahlswede,et al.  Strong converse for identification via quantum channels , 2000, IEEE Trans. Inf. Theory.

[9]  Alexander Russell,et al.  Random Cayley Graphs are Expanders: a Simple Proof of the Alon-Roichman Theorem , 2004, Electron. J. Comb..

[10]  R. Vershynin.  Frame expansions with erasures: an approach through the non-commutative operator theory , 2004, math/0405566.

[11]  S. Mendelson,et al.  On singular values of matrices with independent rows , 2006 .

[12]  Mark Rudelson,et al.  Sampling from large matrices: An approach through geometric functional analysis , 2005, JACM.

[13]  E. Candès,et al.  Sparsity and incoherence in compressive sampling , 2006, math/0611957.

[14]  R. Vershynin.  Spectral norm of products of random and deterministic matrices , 2008, 0812.2432.

[15]  J. Tropp.  On the conditioning of random subdictionaries , 2008 .

[16]  R. Adamczak,et al.  Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles , 2009, 0903.2323.

[17]  G. Grimmett,et al.  ELECTRONIC COMMUNICATIONS in PROBABILITY , 2010 .

[18]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[19]  Nikhil Srivastava,et al.  Graph Sparsification by Effective Resistances , 2011, SIAM J. Comput..