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### Paper:

TR08-103 | 22nd November 2008 00:00

#### Regularity, Boosting, and Efficiently Simulating Every High-Entropy Distribution

TR08-103
Publication: 23rd November 2008 00:06
Keywords:

Abstract:

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.

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