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We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, E H) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in access control, property… (More)

We introduce a framework for proving lower bounds on computational problems over distributions, based on a class of algorithms called <i>statistical algorithms</i>. For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather… (More)

We consider the problem of learning sparse parities in the presence of noise. For learning parities on r out of n variables, we give an algorithm that runs in time poly log 1 δ , 1 1−2η n (1+(2η) 2 +o(1))r/2 and uses only r log(n/δ)ω(1) (1−2η) 2 samples in the random noise setting under the uniform distribution, where η is the noise rate and δ is the… (More)

A basic goal in property testing is to identify a minimal set of features that make a property testable. For the case when the property to be tested is membership in a binary linear error-correcting code, Alon et al. (Trans Inf Theory, 51(11):4032–4039, 2005) had conjectured that the presence of a single low-weight codeword in the dual, and “2-transitivity”… (More)

Motivated by questions in property testing, we search for linear error-correcting codes that have the " single local orbit " property: i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every " sparse " binary code whose coordinates are indexed by elements of F 2 n for… (More)

In the history of property testing, a particularly important role has been played by linear-invariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed-Muller codes, and Fourier sparsity. In this work, we describe a framework… (More)

Affine-invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field F q n to the subfield F q and include all properties that form an F… (More)

Given a pair of finite groups G and H, the set of homomorphisms from G to H form an error-correcting code where codewords differ in at least 1/2 the coordinates. We show that for every pair of <i>abelian</i> groups G and H, the resulting code is (locally) list-decodable from a fraction of errors arbitrarily close to its distance. At the heart of this result… (More)

We develop a framework for proving lower bounds on computational problems over distributions , including optimization and unsupervised learning. Our framework is based on defining a restricted class of algorithms, called statistical algorithms, that instead of accessing samples from the input distribution can only obtain an estimate of the expectation of… (More)

Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. Transitive-closure spanners are a common abstraction for applications in access control, property testing and data structures. We show a… (More)