TR08-103 Authors: Luca Trevisan, Madhur Tulsiani, Salil Vadhan

Publication: 23rd November 2008 00:06

Downloads: 2415

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We show that every high-entropy distribution is indistinguishable from an

efficiently samplable distribution of the same entropy. Specifically, we prove

that if $D$ is a distribution over $\{ 0,1\}^n$ of min-entropy at least $n-k$,

then for every $S$ and $\epsilon$ there is a circuit $C$ of size at most

$S\cdot \poly(\epsilon^{-1}, 2^k)$ that samples a distribution of entropy at

least $n-k$ that is $\epsilon$-indistinguishable from $D$ by circuits of size

$S$.

Stated in a more abstract form (where we refer to indistinguishability by

arbitrary families of distinguishers rather than bounded-size circuits), our

result implies (a) the Weak Szemer\'edi Regularity Lemma of Frieze and Kannan (b) a

constructive version of the Dense Model Theorem of Green, Tao and Ziegler with

better quantitative parameters (polynomial rather than exponential in the

distinguishing probability $\epsilon$), and (c) the Impagliazzo Hardcore Set

Lemma. It appears to be the general result underlying the known connections

between ``regularity'' results in graph theory, ``decomposition'' results in

additive combinatorics, and the Hardcore Lemma in complexity theory.

We present two proofs of our result, one in the spirit of Nisan's proof of

the Hardcore Lemma via duality of linear programming, and one similar to

Impagliazzo's ``boosting'' proof. A third proof by iterative partitioning,

which gives the complexity of the sampler to be exponential in $1/\epsilon$

and $2^k$, is also implicit in the Green-Tao-Ziegler proofs of the Dense Model

Theorem.