News for October 2014

Halloween came early for the property testing community this year, with lots of treats throughout the month of October.

There was FOCS 2014, with a number of relevant contributions including On Learning and Testing Dynamic Environments by Oded Goldreich and Dana Ron (discussed previously here), New algorithms and lower bounds for monotonicity testing by Xi Chen, Rocco Servedio, and Li-Yang Tan and An Automatic Inequality Prover and Instance Optimal Identity Testing by Greg and Paul Valiant.

There were also two workshops on the first day of FOCS with very close connections to property testing. The first, Sparse Fourier Transform: Theory and Applications, was centered around recent developments in sublinear-time algorithms for computing the Fourier transform of signals that have a sparse spectrum (or are close to it). The second, Higher-order Fourier Analysis, discussed recent applications of this area of research in computer science; one of which is in testing algebraic properties of functions. The slides for the talks at these workshops are available online.

Another treat (or a promise of future treats) came in the form of the list of accepted papers for ITCS 2015. As we can see from the list, there will be many presentations at the conference on property testing topics, and we can look forward to lots of interesting reading when these papers are posted online.

Finally, we have this month’s contributions:

Gowers Norm, Function Limits, and Parameter Estimation by Yuichi Yoshida. (arXiv) There has been a lot of recent progress on the problem of characterizing the set of algebraic properties of functions that can be tested with a constant number of queries. This line of work has been notable not just for its success, but also for the connections it has established between property testing and other areas of mathematics (such as higher-order Fourier analysis, for example). In this work, this problem (and generalizations of it) is revisited from a different angle, by considering a new notion of function limits. As the results in the paper show, this notion leads to alternative proofs of the constant-query testability of many classic algebraic properties of functions, and promises to offer a fruitful new line of inquiry.

Testing Identity of Structured Distributions by Ilias Diakonikolas, Daniel M. Kane, and Vladimir Nikishkin. (arXiv) The most fundamental problem in distribution testing is that of testing identity: given sample access to some unknown distribution q, is it identical (or far from) some known distribution p? This problem has been studied extensively and is well-understood: \(\Theta(\sqrt{n})\) queries are both necessary and sufficient to test identity of distributions with support size \(n\). The current paper considers a natural follow-up question: if our distributions p and q are not arbitrary distributions but instead come from some (potentially large but structured) class of distributions, can we test identity more efficiently in this setting? The results of the paper give an emphatic affirmative answer to this question, showing that for many natural classes of distributions, identity can be tested quite efficiently.

Testing Poisson Binomial Distributions by Jayadev Acharya and Constantinos Daskalakis. (arXiv) This paper considers another fundamental problem in distribution testing: how many samples must we draw from some unknown distribution q on \([n]\) to test whether it is a sum of Bernoulli distributions or not? Such distributions are known as Poisson Binomial distributions, and the current paper gives tight upper and lower bounds of \(\Theta(n^{1/4})\) samples for this testing task.

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