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  • Karl Wimmer
  • 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)
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 uncorre-lated 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} n → {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)
The non-linear invariance principle of Mossel, O'Donnell and Oleszkiewicz establishes that if f (x 1 ,. .. , x n) is a multilinear low-degree polynomial with low influences then the distribution of f (B 1 ,. .. , B n) is close (in various senses) to the distribution of f (G 1 ,. .. , G n), where B i ∈ R {−1, 1} are independent Bernoulli random variables and(More)
Recently, the Kahn–Kalai–Linial (KKL) Theorem on influences of functions on {0, 1} n was extended to the setting of functions on Schreier graphs. Specifically, it was shown that for an undirected Schreier graph Sch(G, X, U) with log-Sobolev constant ρ and generating set U closed under conjugation, if f : X → {0, 1} then E[f ] log(1/M[f ]) · ρ · Var[f ].(More)
  • Karl Wimmer
  • 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)