Abstract
We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution D that is δ-dense in a distribution that is ε′-indistinguishable from uniform, there exists a “dense model” for D, that is, a distribution that is δ-dense in the uniform distribution and is ε-indistinguishable from D. This ε-indistinguishability is with respect to an arbitrary small class of functions F. For the natural case where ε′ ≥ Ω(εδ) and ε ≥ δO(1), our lower bound implies that Ω(√(1/ε) log(1/δ)·log|F|) advice bits are necessary for a certain type of reduction that establishes a stronger form of the Dense Model Theorem (and which encompasses all known proofs of the Dense Model Theorem in the literature). There is only a polynomial gap between our lower bound and the best upper bound for this case (due to Zhang), which is O((1/ε2)log(1/δ)·log|F|). Our lower bound can be viewed as an analogue of list size lower bounds for list-decoding of error-correcting codes, but for “dense model decoding” instead.
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Index Terms
Advice Lower Bounds for the Dense Model Theorem
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