81,238 research outputs found
Two Structural Results for Low Degree Polynomials and Applications
In this paper, two structural results concerning low degree polynomials over
finite fields are given. The first states that over any finite field
, for any polynomial on variables with degree , there exists a subspace of with dimension on which is constant. This result is shown to be tight.
Stated differently, a degree polynomial cannot compute an affine disperser
for dimension smaller than . Using a recursive
argument, we obtain our second structural result, showing that any degree
polynomial induces a partition of to affine subspaces of dimension
, such that is constant on each part.
We extend both structural results to more than one polynomial. We further
prove an analog of the first structural result to sparse polynomials (with no
restriction on the degree) and to functions that are close to low degree
polynomials. We also consider the algorithmic aspect of the two structural
results.
Our structural results have various applications, two of which are:
* Dvir [CC 2012] introduced the notion of extractors for varieties, and gave
explicit constructions of such extractors over large fields. We show that over
any finite field, any affine extractor is also an extractor for varieties with
related parameters. Our reduction also holds for dispersers, and we conclude
that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over
.
* Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine
disperser over a prime field is also an affine extractor with related
parameters. Using our structural results, and based on the work of Kaufman and
Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this
result to any constant degree
Learning pseudo-Boolean k-DNF and Submodular Functions
We prove that any submodular function f: {0,1}^n -> {0,1,...,k} can be
represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a
natural generalization of DNF representation for functions with integer range.
Each term in such a formula has an associated integral constant. We show that
an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all
constants associated with the terms of the formula are bounded.
This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to
pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership
queries under the uniform distribution for submodular functions of the form
f:{0,1}^n -> {0,1,...,k}. Our algorithm runs in time polynomial in n, k^{O(k
\log k / \epsilon)}, 1/\epsilon and log(1/\delta) and works even in the
agnostic setting. The line of previous work on learning submodular functions
[Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi,
Klivans, Kothari, Lee (SODA '12)] implies only n^{O(k)} query complexity for
learning submodular functions in this setting, for fixed epsilon and delta.
Our learning algorithm implies a property tester for submodularity of
functions f:{0,1}^n -> {0, ..., k} with query complexity polynomial in n for
k=O((\log n/ \loglog n)^{1/2}) and constant proximity parameter \epsilon
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
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