17 research outputs found

    A Small PRG for Polynomial Threshold Functions of Gaussians

    Full text link
    We develop a pseudo-random generator to fool degree-dd polynomial threshold functions with respect to the Gaussian distribution. For c>0c>0 any constant, we construct a pseudo-random generator that fools such functions to within ϵ\epsilon and has seed length log(n)2O(d)ϵ4c\log(n) 2^{O(d)} \epsilon^{-4-c}

    A Pseudorandom Generator for Polynomial Threshold Functions of Gaussian with Subpolynomial Seed Length

    Get PDF
    We develop a pseudorandom generator that fools degree-dd polynomial threshold functions in nn variables with respect to the Gaussian distribution and has seed length Oc,d(log(n)ϵc)O_{c,d}(\log(n) \epsilon^{-c})

    A Polylogarithmic PRG for Degree 22 Threshold Functions in the Gaussian Setting

    Get PDF
    We devise a new pseudorandom generator against degree 2 polynomial threshold functions in the Gaussian setting. We manage to achieve ϵ\epsilon error with seed length polylogarithmic in ϵ\epsilon and the dimension, and exponential improvement over previous constructions

    The Correct Exponent for the Gotsman-Linial Conjecture

    Get PDF
    We prove a new bound on the average sensitivity of polynomial threshold functions. In particular we show that a polynomial threshold function of degree dd in at most nn variables has average sensitivity at most n(log(n))O(dlog(d))2O(d2log(d)\sqrt{n}(\log(n))^{O(d\log(d))}2^{O(d^2\log(d)}. For fixed dd the exponent in terms of nn 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 Θ(dn)\Theta(d\sqrt{n})

    A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut

    Full text link
    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)

    Deterministic Approximate Counting of Polynomial Threshold Functions via a Derandomized Regularity Lemma

    Get PDF

    Pseudorandomness via the discrete Fourier transform

    Full text link
    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

    Full text link
    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 ϵ\epsilon and has an almost optimal seed-length of O(logn+log(1/ϵ)loglog(1/ϵ))O(\log n + \log(1/\epsilon) \cdot \log\log(1/\epsilon)). For an inverse-polynomially growing error ϵ\epsilon, our generator has a seed-length optimal up to a factor of O(loglog(n))O( \log \log {(n)}). The most efficient PRG previously known (due to Kane, 2012) requires a seed-length of Ω(log3/2(n))\Omega(\log^{3/2}{(n)}) 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
    corecore