Noncommutative Bennett and Rosenthal inequalities

Abstract

In this paper we extend the Bernstein, Prohorov and Bennett inequalities to the noncommutative setting. In addition we provide an improved version of the noncommutative Rosenthal inequality, essentially due to Nagaev, Pinelis and Pinelis, Utev for commutative random variables. We also present new best constants in Rosenthal's inequality. Applying these results to random Fourier projections, we recover and elaborate on fundamental results from compressed sensing, due to Candes, Romberg and Tao.

arXiv commentPublished in at http://dx.doi.org/10.1214/12-AOP771 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Recovering low-rank matrices from few coefficients in any basis
• Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps
• Non-commutative martingale inequalities
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• Subadditivity of Matrix phi-Entropy and Concentration of Random Matrices
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• Noncommutative martingale deviation and Poincaré type inequalities with applications
• Subgaussian 1-cocycles on discrete groups
• Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing
• Poincaré type inequalities for group measure spaces and related transportation cost inequalities
• Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem
• Smooth Fourier multipliers on group von Neumann algebras
• Sharp Rosenthal-type inequalities for mixtures and log-concave variables
• Ultraproduct methods for mixed $q$-Gaussian algebras
• A note on quantum expanders
• Concentration inequalities using higher moments information
• Noncommutative versions of the arithmetic-geometric mean inequality
• Moment inequalities for matrix-valued U-statistics of order 2
• Concentration and moment inequalities for sums of independent heavy-tailed random matrices
• Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I
• Positivity of state, trace, and moment polynomials, and applications in quantum information