Clément L. Canonne

Learning Circuits with few Negations.

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).

Testing Probability Distributions Underlying Aggregated Data

In this paper, we analyze and study a hybrid model for testing and learning probability distributions. Here, in addition to samples, the testing algorithm is provided with one of two different types of oracles to the unknown distribution D over [n]. More precisely, we consider both the dual and cumulative dual access models, in which the algorithm A can both sample from D and respectively, for any i ∈ [n], query the probability mass D(i) (query access); or get the total mass of {1,…,i}, i.e. \(\sum_{j=1}^i D(j)\) (cumulative access)

Are Few Bins Enough: Testing Histogram Distributions

A probability distribution over an ordered universe [n]={1,...,n} is said to be a k-histogram if it can be represented as a piecewise-constant function over at most k contiguous intervals. We study the following question: given samples from an arbitrary distribution D over [n], one must decide whether D is a k-histogram, or is far in L_1 distance from any such succinct representation. We obtain a sample and time-efficient algorithm for this problem, complemented by a nearly-matching information-theoretic lower bound on the number of samples required for this task. Our results significantly improve on the previous state-of-the-art, due to Indyk, Levi, and Rubinfeld 2012) and Canonne, Diakonikolas, Gouleakis, and Rubinfeld (2016).

A Chasm Between Identity and Equivalence Testing with Conditional Queries.

A recent model for property testing of probability distributions (Chakraborty et al., ITCS 2013, Canonne et al., SICOMP 2015) enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain. In particular, Canonne, Ron, and Servedio (SICOMP 2015) showed that, in this setting, testing identity of an unknown distribution

An adaptivity hierarchy theorem for property testing

D

Distribution Testing Lower Bounds via Reductions from Communication Complexity

(whether

Adaptive estimation in weighted group testing

D=D^\ast

Generalized Uniformity Testing

for an explicitly known

Testing conditional independence of discrete distributions

D^\ast

Testing Shape Restrictions of Discrete Distributions

) can be done with a constant number of queries, independent of the support size