Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications

Ahlswede and Winter (IEEE Trans. Inf. Th. 2002) introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan (JCSS 1988)). As a consequence, we derandomize an efficient construction by Alon and Roichman (RSA 1994) of an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson (STOC 2004). We also apply these pessimistic estimators to the problem of solving semidefinite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of Ahslwede and Winter.

[1]  Arya Mazumdar,et al.  Codes on hypergraphs , 2008, 2008 IEEE International Symposium on Information Theory.

[2]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[3]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[4]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[5]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[6]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

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

[8]  N. Z. Shor.  Cut-off method with space extension in convex programming problems , 1977, Cybernetics.

[9]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[10]  Shlomo Hoory,et al.  On codes from hypergraphs , 2004, Eur. J. Comb..

[11]  Oded Goldreich,et al.  A Sample of Samplers - A Computational Perspective on Sampling (survey) , 1997, Electron. Colloquium Comput. Complex..

[12]  D. Spielman,et al.  Computationally efficient error-correcting codes and holographic proofs , 1995 .

[13]  Avi Wigderson,et al.  Derandomizing homomorphism testing in general groups , 2004, STOC '04.

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

[15]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[16]  Avi Wigderson,et al.  A randomness-efficient sampler for matrix-valued functions and applications , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[17]  Oded Goldreich,et al.  Locally testable codes and PCPs of almost-linear length , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[18]  Oded Goldreich,et al.  Locally testable codes and PCPs of almost-linear length , 2006, JACM.

[19]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[20]  Joel H. Spencer,et al.  Integral approximation sequences , 1984, Math. Program..

[21]  Stavros G. Kolliopoulos,et al.  Approximation Algorithms for Covering/Packing Integer Programs , 2002, cs/0205030.

[22]  Leonard J. Schulman,et al.  Improved Expansion of Random Cayley Graphs , 2004, Discret. Math. Theor. Comput. Sci..

[23]  Russell Impagliazzo,et al.  How to recycle random bits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[24]  Leonid A. Levin,et al.  Security preserving amplification of hardness , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[25]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[26]  Noam Nisan,et al.  Extracting Randomness: A Survey and New Constructions , 1999, J. Comput. Syst. Sci..

[27]  Sanjeev Arora,et al.  A combinatorial, primal-dual approach to semidefinite programs , 2007, STOC '07.

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

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

[30]  Noga Alon,et al.  Derandomized graph products , 1995, computational complexity.

[31]  Noga Alon,et al.  Random Cayley Graphs and Expanders , 1994, Random Struct. Algorithms.