Showing papers by "Eric Blais published in 2014"

Lower Bounds for Testing Properties of Functions over Hypergrid Domains

Abstract: We show how the communication complexity method introduced in (Blais, Brody, Matulef 2012) can be used to prove lower bounds on the number of queries required to test properties of functions with non-hypercube domains. We use this method to prove strong, and in many cases optimal, lower bounds on the query complexity of testing fundamental properties of functions over hypergrid domains: monotonicity, the Lipschitz property, separate convexity, convexity and monotonicity of higher-order derivatives. There is a long line of work on upper bounds and lower bounds for many of these properties that uses a diverse set of combinatorial techniques. Our method provides a unified treatment of lower bounds for all these properties based on Fourier analysis. A key ingredient in our new lower bounds is a set of Walsh functions, a canonical Fourier basis for the set of functions on the line. The orthogonality of the Walsh functions lets us use a product construction to extend our method from properties of functions over the line to properties of functions over hypergrids. Our product construction applies to properties over hypergrids that can be expressed in terms of axis-parallel directional derivatives, such as monotonicity, the Lipschitz property and separate convexity. We illustrate the robustness of our method by making it work for convexity, which is the property of the Hessian matrix of second derivatives being positive semidefinite and thus cannot be described by axis-parallel directional derivatives alone. Such robustness contrasts with the state of the art in the upper bounds for testing properties over hypergrids: methods that work for other properties are not applicable for testing convexity, for which no nontrivial upper bounds are known for d ≥ 2.

The Information Complexity Of Hamming Distance

Abstract: The Hamming distance function Ham_{n,d} returns 1 on all pairs of inputs x and y that differ in at most d coordinates and returns 0 otherwise. We initiate the study of the information complexity of the Hamming distance function. We give a new optimal lower bound for the information complexity of the Ham_{n,d} function in the small-error regime where the protocol is required to err with probability at most epsilon < d/n. We also give a new conditional lower bound for the information complexity of Ham_{n,d} that is optimal in all regimes. These results imply the first new lower bounds on the communication complexity of the Hamming distance function for the shared randomness two-way communication model since Pang and El-Gamal (1986). These results also imply new lower bounds in the areas of property testing and parity decision tree complexity.

Semi-Strong Colouring of Intersecting Hypergraphs

Abstract: For any c ≥ 2, a c-strong colouring of the hypergraph G is an assignment of colours to the vertices of G such that, for every edge e of G, the vertices of e are coloured by at least min{c,|e|} distinct colours. The hypergraph G is t-intersecting if every two edges of G have at least t vertices in common.A natural variant of a question of Erdős and Lovasz is: For fixed c ≥ 2 and t ≥ 1, what is the minimum number of colours that is sufficient to c-strong colour any t-intersecting hypergraphs? The purpose of this note is to describe some open problems related to this question.

On DNF Approximators for Monotone Boolean Functions

Abstract: We study the complexity of approximating monotone Boolean functions with disjunctive normal form (DNF) formulas, exploring two main directions. First, we construct DNF approximators for arbitrary monotone functions achieving one-sided error: we show that every monotone f can be e-approximated by a DNF g of size \(2^{n-\Omega_\epsilon(\sqrt{n})}\) satisfying g(x) ≤ f(x) for all x ∈ {0,1} n . This is the first non-trivial universal upper bound even for DNF approximators incurring two-sided error.

Learning circuits with few negations

Abstract: Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A. A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).

Rapid Sampling for Visualizations with Ordering Guarantees

Abstract: Visualizations are frequently used as a means to understand trends and gather insights from datasets, but often take a long time to generate. In this paper, we focus on the problem of rapidly generating approximate visualizations while preserving crucial visual proper- ties of interest to analysts. Our primary focus will be on sampling algorithms that preserve the visual property of ordering; our techniques will also apply to some other visual properties. For instance, our algorithms can be used to generate an approximate visualization of a bar chart very rapidly, where the comparisons between any two bars are correct. We formally show that our sampling algorithms are generally applicable and provably optimal in theory, in that they do not take more samples than necessary to generate the visualizations with ordering guarantees. They also work well in practice, correctly ordering output groups while taking orders of magnitude fewer samples and much less time than conventional sampling schemes.

Learning circuits with few negations.

Abstract: Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).