This month saw more development on testing properties of distributions and a result with intriguing connections to property testing. And for readers who may have missed it, Clément Canonne and Gautam Kamath wrote an engaging survey of some recent work on testing properties of distribution here.

Optimal algorithms and lower bounds for testing closeness of structured distributions by Ilias Diakonikolas, Daniel M. Kane, and Vladimir Nikishkin (arXiv). One of the fundamental results in testing properties of distributions is that if we want to estimate the (\(L_1\)) distance between two unknown distributions on a domain of size \(n\) up to some constant additive factor,  we need to draw \(O(n^{2/3})\) samples from these distributions, and this sample complexity is tight in general. But what if we consider the same problem in the setting where we are promised that the distributions come from some (known) class of distribution? This paper shows that, for many natural classes of distributions, we can obtain much more efficient algorithms for testing the closeness of distributions in this setting.

Smooth Boolean functions are easy: efficient algorithms for low-sensitivity functions by Parikshit Gopalan, Noam Nisan, Rocco A. Servedio, Kunal Talwar, and Avi Wigderson (arXiv). The (maximum) sensitivity of a Boolean function \(f : \{0,1\}^n \to \{0,1\}\) is the size of the largest set \(S \subseteq [n]\) such that there is an input \(x \in \{0,1\}^n\) for which \(f(x) = f(y)\) for every input \(y\) obtained by flipping the value of the \(i\)th coordinate of \(x\) for some \(i \in S\). One of the results in this paper shows that functions with low sensitivity can be locally self-corrected: given query access to a function \(r : \{0,1\}^n \to \{0,1\}\) that is close to a function \(f\) with low sensitivity (think of \(r\) as a corrupted version of \(f\)), there is an algorithm that for any input $x \in \{0,1\}^n$, queries \(r\) on a few inputs and outputs with high probability the value \(f(x)\). This notion of local self-correction is of course closely related to locally-decodable codes; it is also one of the fundamental techniques used to obtain many results in property testing as well (see for example Chapter 3 of Dana Ron’s survey). Can this result, or the techniques used to obtain it, also yield new results in property testing?