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We generalize the Kahn-Kalai-Linial (KKL) Theorem to random walks on Cayley and Schreier graphs, making progress on an open problem of Hoory, Linial, and Wigderson. In our generalization, the underlying group need not be abelian so long as the generating set is a union of conjugacy classes.
  • Karl Wimmer
  • 2014 IEEE 29th Conference on Computational…
  • 2014
One of the classic results in analysis of Boolean functions is a result of Friedgut~cite{Fri98} that states that Boolean functions over the hypercube of low influence are approximately juntas, functions which are determined by few coordinates. While this result has also been extended to product distributions, not much is known in the case of nonproduct(More)
We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We give the first efficient algorithms for testing whether a Boolean function has a sparse(More)
Say that f : {0, 1}n → {0, 1} 2-approximates g : {0, 1}n → {0, 1} if the functions disagree on at most an 2 fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits: (1) For every constant 0 < 2 < 1/2 there is a DNF of size 2 √ n) that 2-approximates the Majority function on n bits, and this is optimal up(More)
In recent work, Kalai, Klivans, Mansour, and Servedio (2005) studied a variant of the “Low-Degree (Fourier) Algorithm” for learning under the uniform probability distribution on {0,1} n . They showed that the L 1 polynomial regression algorithm yields agnostic (tolerant to arbitrary noise) learning algorithms with respect to the class of threshold(More)
A function f is d-resilient if all its Fourier coefficients of degree at most d are zero, i.e. f is uncorrelated with all low-degree parities. We study the notion of approximate resilience of Boolean functions, where we say that f is α-approximately d-resilient if f is α-close to a [−1, 1]-valued d-resilient function in `1 distance. We show that approximate(More)
We study lower bounds for testing membership in families of linear/affine-invariant Boolean functions over the hypercube. A family of functions P ⊆ {{0, 1} → {0, 1}} is linear/affine invariant if for any f ∈ P , it is the case that f ◦L ∈ P for any linear/affine transformation L of the domain. Motivated by the recent resurgence of attention to the(More)
  • Karl Wimmer
  • 2010 IEEE 51st Annual Symposium on Foundations of…
  • 2010
We generalize algorithms from computational learning theory that are successful under the uniform distribution on the Boolean hypercube $\{0,1\}^n$ to algorithms successful on permutation invariant distributions. A permutation invariant distribution is a distribution where the probability mass remains constant upon permutations in the instances. While the(More)
The non-linear invariance principle of Mossel, O’Donnell and Oleszkiewicz establishes that if fpx1, . . . , xnq is a multilinear low-degree polynomial with low influences then the distribution of fpB1, . . . ,Bnq is close (in various senses) to the distribution of fpG1, . . . ,Gnq, where Bi PR t ́1, 1u are independent Bernoulli random variables and Gi „(More)
In a very strong positive result for passive learning algorithms, Bshouty et al. showed that DNF expressions are efficiently learnable in the uniform random walk model. It is natural to ask whether the more expressive class of thresholds of parities (TOP) can also be learned efficiently in this model, since both DNF and TOP are efficiently uniform-learnable(More)