March was a relatively good month for property testing, with 3 papers — including a foray in differential privacy.

**Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps**, by Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri (arXiv). Testing *unateness* (that is, whether a function is monotone after a rotation of the hypercube) has recently received significant attention. Understanding the power of adaptivity in property testing, on the other hand, has been a long-standing question, which has sparked much interest over the past decade. In this paper, the authors obtain optimal results for testing unateness of real-valued functions \(f\colon \{0,1\}^d\to \mathbb{R}\) and \(f\colon [n]^d\to \mathbb{R}\) (up to the dependence on the proximty parameter \(\varepsilon\)), and show that — unlike what happens for monotonicity testing of real-valued functions — *adaptivity helps*. Namely, adaptive testing of unatess of functions \(f\colon \{0,1\}^d\to \mathbb{R}\) has query complexity \(O(d/\varepsilon)\), but non-adaptive testers require \(\Omega(d\log d)\) queries.

*This work is a merged, and significantly improved version of two papers we covered a few months back, by the same authors.*

Switching from function to distribution testing, last month saw two independent samples appear online:

**Near-Optimal Closeness Testing of Discrete Histogram Distributions**, by Ilias Diakonikolas, Daniel M. Kane, and Vladimir Nikishkin (arXiv). Closeness testing of discrete distributions is the question of testing, given sample access to two unknown distributions \(p,q\) over a known discrete domain \([n] \stackrel{\rm def}{=} \{1,\dots,n\}\), whether \(p=q\) or \(d_{\rm TV}(p,q)> \varepsilon\) (are the two distributions the same, or differ significantly in total variation distance?). This quite fundamental question is by now fully understood in the general case, where \(p,q\) are allowed to be *arbitrary* distributions over ; *but what if one has some additional knowledge about them?* Specifically, what if we were guaranteed that both distributions have some “nice structure” — e.g., that they are piecewise constant with \(k\) pieces (\(k\)-histograms)? Leveraging machinery the authors developed in a series of previous work (for the continuous case), the authors show that in this case, the sample complexity of “testing closeness under structural assumptions” is quite different than in the general case; and that the resulting (near tight) bound is a rather intricate tradeoff between the three parameters \(n,k,\varepsilon\).

**Priv’IT: Private and Sample Efficient Identity Testing,** by Bryan Cai, Constantinos Daskalakis, and Gautam Kamath (arXiv). Distribution testing is a vibrant field; differential privacy (DP) is an incredibly active and topical area. What about *differentially private distribution testing* — a.k.a. testing the data without learning too much about any single sample? In this paper, the authors address the task of performing identity testing of distributions (“given samples from an unknown arbitrary distribution, decide whether it matches a fixed known probability distribution/model”) in the DP setting, focusing on getting the best of both worlds: how to guarantee privacy without sacrificing efficiency. Not only do they obtain asymptotically near-optimal bounds in both the testing and differential privacy parameters — they also run some experiments to validate their approach empirically. (Plus, the title is rather cute.)

*Usual disclaimer: if we forgot or misrepresented your paper, please signal it in the comments — we’ll address it as quickly as we can.*