Showing papers by "Eric Blais published in 2015"

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 properties 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.

Performance Prediction of Configurable Software Systems by Fourier Learning (T)

Abstract: Understanding how performance varies across a large number of variants of a configurable software system is important for helping stakeholders to choose a desirable variant. Given a software system with n optional features, measuring all its 2^n possible configurations to determine their performances is usually infeasible. Thus, various techniques have been proposed to predict software performances based on a small sample of measured configurations. We propose a novel algorithm based on Fourier transform that is able to make predictions of any configurable software system with theoretical guarantees of accuracy and confidence level specified by the user, while using minimum number of samples up to a constant factor. Empirical results on the case studies constructed from real-world configurable systems demonstrate the effectiveness of our algorithm.

Quantum Algorithm for Monotonicity Testing on the Hypercube

Abstract: In this note, we develop a bounded-error quantum algorithm that makes ˜ O(n 1=4 e 1=2 ) queries to a function f :f0; 1g n !f0; 1g, accepts when f is monotone, and rejects when f is e-far from being monotone. This result gives a super-quadratic improve- ment compared to the best known randomized algorithm for all e = o(1). The improvement is cubic when e = 1= p n.

Partially Symmetric Functions Are Efficiently Isomorphism Testable

Abstract: Given a Boolean function $f$, the $f$-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to $f$ up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions $f$ we can test $f$-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main resul...

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).

Approximating Boolean Functions with Depth-2 Circuits

Abstract: We study the complexity of approximating Boolean functions with disjunctive normal forms (DNFs) and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function. In the first direction, our main positive results are the first nontrivial universal upper bounds on approximability by DNFs: (a) every Boolean function can be $\epsilon$-approximated by a DNF of size $O_\epsilon(2^n/\log n)$, and (b) every Boolean function can be $\epsilon$-approximated by a DNF of width $c_\epsilon\, n$, where $c_\epsilon < 1$. Our techniques extend broadly to give strong universal upper bounds on approximability by various depth-2 circuits that generalize DNFs, including the intersection of halfspaces, low-degree Polynomial threshold functions, and unate functions. We show that the parameters of our constructions almost match the information-theoretic inapproximability of a random function. In the second direction our mai...

An inequality for the Fourier spectrum of parity decision trees

Abstract: We give a new bound on the sum of the linear Fourier coefficients of a Boolean function in terms of its parity decision tree complexity. This result generalizes an inequality of O'Donnell and Servedio for regular decision trees. We use this bound to obtain the first non-trivial lower bound on the parity decision tree complexity of the recursive majority function.

A Polynomial Lower Bound for Testing Monotonicity

Abstract: We show that every algorithm for testing $n$-variate Boolean functions for monotonicity must have query complexity $\tilde{\Omega}(n^{1/4})$. All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only $\Omega(\log n)$. Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015) recently showed that non-adaptive algorithms require almost $\Omega(n^{1/2})$ queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity $O(\log n)$ when the input is a regular LTF.

Partially Symmetric Functions Are Efficiently Isomorphism Testable

Abstract: Given a Boolean function $f$, the $f$-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to $f$ up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions $f$ we can test $f$-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main resul...

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 properties 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.

Quantum Algorithm for Monotonicity Testing on the Hypercube

Abstract: In this note, we develop a bounded-error quantum algorithm that makes $\tilde O(n^{1/4}\varepsilon^{-1/2})$ queries to a Boolean function $f$, accepts a monotone function, and rejects a function that is $\varepsilon$-far from being monotone. This gives a super-quadratic improvement compared to the best known randomized algorithm for all $\varepsilon = o(1)$. The improvement is cubic when $\varepsilon = 1/\sqrt{n}$.