Abstract. 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 first give an efficient algorithm for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of F n 2 (equivalently, for testing whether f is a junta over a small number of parities). We next give an efficient algorithm for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients). In both cases we also prove lower bounds showing that any testing algorithm -even an adaptive onemust have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Finally, we give an "implicit learning" algorithm that lets us test any sub-property of Fourier concision. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [13].

The non-linear invariance principle of Mossel, O'Donnell and Oleszkiewicz establishes that if f px1, .

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 functions-under certain restricted instance distributions, including uniform on {0, 1} n and Gaussian on R n . In this work we show how all learning results based on the Low-Degree Algorithm can be generalized to give almost identical agnostic guarantees under arbitrary product distributions on instance spaces X 1 × · · · × X n . We also extend these results to learning under mixtures of product distributions.The main technical innovation is the use of (Hoeffding) orthogonal decomposition and the extension of the "noise sensitivity method" to arbitrary product spaces. In particular, we give a very simple proof that threshold functions over arbitrary product spaces have δ-noise sensitivity O ( √ δ), resolving an open problem suggested by Peres (2004).

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Abstract. 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 spectral norm of g. That is, if g is close to having spectral norm s, then we can test g-isomorphism with poly(s) queries, and if g is far from having spectral norm s, then we cannot test g-isomorphism with o(log s) queries. The upper bound is almost tight since there is indeed a function g close to having spectral norm s whereas testing gisomorphism requires Ω(s) queries. As far as we know, our result is the first characterization of this type for functions. Our upper bound is essentially the Kushilevitz-Mansour learning algorithm, modified for use in the implicit setting. Exploiting our upper bound, we show that any property is testable if it can be well-approximated by functions with small spectral norm. We also extend our algorithm to the setting where A is allowed to be singular.

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 ℓ 1 distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class C over the uniform distribution. Roughly speaking, if all functions in a class C are far from being d-resilient then C can be learned agnostically in time n O(d) and conversely, if C contains a function close to being d-resilient then agnostic learning of C in the statistical query (SQ) framework of Kearns has complexity of at least n Ω(d) . This characterization is based on the duality between ℓ 1 approximation by degree-d polynomials and approximate d-resilience that we establish. In particular, it implies that ℓ 1 approximation by low-degree polynomials, known to be sufficient for agnostic learning over product distributions, is in fact necessary.Focusing on monotone Boolean functions, we exhibit the existence of near-optimal α-approximately Ω(α √ n)-resilient monotone functions for all α > 0. Prior to our work, it was conceivable even that every monotone function is Ω(1)-far from any 1-resilient function. Furthermore, we construct simple, explicit monotone functions based on Tribes and CycleRun that are close to highly resilient functions. Our constructions are based on general resilience analysis and amplification techniques we introduce. These structural results, together with the characterization, imply nearly optimal lower bounds for agnostic learning of monotone juntas, a natural variant of the well-studied junta learning problem. In particular we show that no SQ algorithm can efficiently agnostically learn monotone k-juntas for any k = ω(1) and any constant error less than 1/2.

Abstract. Given an implicit n×n matrix A with oracle access x T Ax for any x ∈ R n , we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum physics, machine learning, and pattern matching. Two metrics are commonly used for evaluating the estimators: i) variance; ii) a high probability multiplicative-approximation guarantee. Almost all the known estimators are of the form 1Our main results are summarized as follows:1. We give an exact characterization of the minimum variance unbiased estimator in the broad class of linear nonadaptive estimators (which subsumes all the existing known estimators). 2. We also consider the query complexity lower bounds for any (possibly nonlinear and adaptive) estimators: (a) We show that any estimator requires Ω(1/ǫ) queries to have a guarantee of variance at most ǫ. (b) We show that any estimator requires Ω( 1 ǫ 2 log 1 δ ) queries to achieve a (1 ± ǫ)-multiplicative approximation guarantee with probability at least 1 − δ. Both above lower bounds are asymptotically tight. As a corollary, we also resolve a conjecture in the seminal work of Avron and Toledo (Journal of the ACM 2011) regarding the sample complexity of the Gaussian Estimator.

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(