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TR06-105 | 23rd August 2006 00:00
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#### Derandomizing the AW matrix-valued Chernoff bound using pessimistic estimators and applications

**Abstract:**
Ahlswede and Winter 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). As a consequence, we derandomize a construction of Alon and Roichman (see also the alternate proof given by Landau-Russell and Loh-Schulman) 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. 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 Ahslwede and Winter.

The results above appear as theorems in the paper "A Randomness-Efficient Sampler for Matrix-Valued Functions and Applications" by the authors, 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). That main theorem stating that the expander walk sampler is good for matrix-valued functions thus remains open. One purpose of the current paper is to show that the applications in that paper hold true despite our inability to prove the expander walk sampler theorem for matrix-valued functions.