Derandomizing the AW matrix-valued Chernoff bound using pessimistic estimators and applications

Ahlswede and Winter [AW02] 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 [Rag88]). As a consequence, we derandomize a construction of Alon and Roichman [AR94] (see also [LR04, LS04]) to efficiently construct an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This also gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson [SW04]. We also apply these pessimistic estimators to the problem of solving semi-definite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of [AW02]. The results above appear as theorems in the paper [WX05a] (see also [WX05b]), as consequences to the main theorem of that paper: a randomness efficient sampler for matrix valued f...

• Type: Journal
• Year: 2006
• Where: ECCC
• Authors: Avi Wigderson, David Xiao
• Keywords: ECCC 2006, Homomorphism Testing Problem, Main Theorem, Pessimistic Estimators