A Matrix Expander Chernoff Bound

We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an expander, confirming a conjecture due to Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the Golden-Thompson inequality which improves in some ways the inequality of Sutter, Berta, and Tomamichel, and may be of independent interest, as well as an adaptation of an argument for the scalar case due to Healy. Secondarily, we also provide a generic reduction showing that any concentration inequality for vector-valued martingales implies a concentration inequality for the corresponding expander walk, with a weakening of parameters proportional to the squared mixing time.Comment: Fixed a minor bug in the proof of Theorem 3.

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], as consequences to the main theorem of that paper: a randomness efficient sampler for matrix valued functions via expander walks. However, we discovered an error in the proof of that main theorem (which we briefly describe in the appendix). One purpose of the current paper is to show that the applications in that paper hold true despite this error.

Retraction of Wigderson-Xiao: A randomness-efficient sampler for matrix valued functions, eccc tr05-107. 2006. [ECCC:TR05-107]. 3

We discovered an error in the proof of the main theorem of A Randomness-Efficient Sampler for Matrix-valued Functions and Applications, which appears as ECCC TR05-107 [WX05a], and also appeared in FOCS [WX05b]. We describe it below. This error invalidates all of the results concerning the expander walk sampler for matrix-valued functions. Nevertheless, we are able to use a different technique to prove the main applications of the sampler that appear in that paper. These include a deterministic algorithm for constructing logarithmicdegree Cayley graphs on any group (derandomizing Alon-Roichman’s theorem [AR94]), and for the quantum hypergraph cover problem, described in Ahlswede-Winter [AW02]. We state the correct result- the manuscript with their proof, under the title Derandomizing the AW matrix-valued Chernoff bound using pessimistic estimators and applications, is available as ECCC TR06-105 [WX06] and also will appear on our homepages 1 Error in the proof of the main theorem We discovered a (seemingly fatal) error in the proof of the main theorem of [WX05a]. This invalidates all the expander walk sampler results of [WX05a]. Fortunately the main applications survive via