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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. An example corollary is that for every f : [n] k → {0, 1} with E[f… (More)

One of the classic results in analysis of Boolean functions is a result of Friedgut [Fri98] that states that Boolean functions over the hypercube of low influence are essentially determined by few coordinates, called juntas. While this result has also been extended to product distributions, not much is known in the case of nonproduct distributions. We… (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 Serve-dio [KKMS05] 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)

Say that f : {0, 1} n → {0, 1}-approximates g : {0, 1} n → {0, 1} if the functions disagree on at most an fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits: (1) For every constant 0 < < 1/2 there is a DNF of size 2 O(√ n) that-approximates the Majority function on n bits, and this is optimal up to… (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)

- Karl Wimmer
- FOCS
- 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 , distributions where the probability mass remains constant upon permutations in the instances. While the tools in our generalization mimic those used… (More)

A function f : F n 2 → {−1, 1} is called linear-isomorphic to g if f = g • A for some non-singular matrix A. In the g-isomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linear-isomorphic to g or far from being so. We show that the query complexity to test g-isomorphism is essentially determined by the… (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)