17 research outputs found
A Small PRG for Polynomial Threshold Functions of Gaussians
We develop a pseudo-random generator to fool degree- polynomial threshold
functions with respect to the Gaussian distribution. For any constant, we
construct a pseudo-random generator that fools such functions to within
and has seed length
A Pseudorandom Generator for Polynomial Threshold Functions of Gaussian with Subpolynomial Seed Length
We develop a pseudorandom generator that fools degree- polynomial
threshold functions in variables with respect to the Gaussian distribution
and has seed length
A Polylogarithmic PRG for Degree Threshold Functions in the Gaussian Setting
We devise a new pseudorandom generator against degree 2 polynomial threshold
functions in the Gaussian setting. We manage to achieve error with
seed length polylogarithmic in and the dimension, and exponential
improvement over previous constructions
The Correct Exponent for the Gotsman-Linial Conjecture
We prove a new bound on the average sensitivity of polynomial threshold
functions. In particular we show that a polynomial threshold function of degree
in at most variables has average sensitivity at most
. For fixed the exponent
in terms of in this bound is known to be optimal. This bound makes
significant progress towards the Gotsman-Linial Conjecture which would put the
correct bound at
A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut
We give improved pseudorandom generators (PRGs) for Lipschitz functions of
low-degree polynomials over the hypercube. These are functions of the form
psi(P(x)), where P is a low-degree polynomial and psi is a function with small
Lipschitz constant. PRGs for smooth functions of low-degree polynomials have
received a lot of attention recently and play an important role in constructing
PRGs for the natural class of polynomial threshold functions. In spite of the
recent progress, no nontrivial PRGs were known for fooling Lipschitz functions
of degree O(log n) polynomials even for constant error rate. In this work, we
give the first such generator obtaining a seed-length of (log
n)\tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps.
Previous generators had an exponential dependence on the degree.
We use our PRG to get better integrality gap instances for sparsest cut, a
fundamental problem in graph theory with many applications in graph
optimization. We give an instance of uniform sparsest cut for which a powerful
semi-definite relaxation (SDP) first introduced by Goemans and Linial and
studied in the seminal work of Arora, Rao and Vazirani has an integrality gap
of exp(\Omega((log log n)^{1/2})). Understanding the performance of the
Goemans-Linial SDP for uniform sparsest cut is an important open problem in
approximation algorithms and metric embeddings and our work gives a
near-exponential improvement over previous lower bounds which achieved a gap of
\Omega(log log n)
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
Almost Optimal Pseudorandom Generators for Spherical Caps
Halfspaces or linear threshold functions are widely studied in complexity
theory, learning theory and algorithm design. In this work we study the natural
problem of constructing pseudorandom generators (PRGs) for halfspaces over the
sphere, aka spherical caps, which besides being interesting and basic geometric
objects, also arise frequently in the analysis of various randomized algorithms
(e.g., randomized rounding). We give an explicit PRG which fools spherical caps
within error and has an almost optimal seed-length of . For an inverse-polynomially
growing error , our generator has a seed-length optimal up to a
factor of . The most efficient PRG previously known (due
to Kane, 2012) requires a seed-length of in this
setting. We also obtain similar constructions to fool halfspaces with respect
to the Gaussian distribution.
Our construction and analysis are significantly different from previous works
on PRGs for halfspaces and build on the iterative dimension reduction ideas of
Kane et. al. (2011) and Celis et. al. (2013), the \emph{classical moment
problem} from probability theory and explicit constructions of \emph{orthogonal
designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion
in Lie groups.Comment: 28 Pages (including the title page