April is now behind us, and we hope you and your families are all staying safe and healthy. We saw ~~six~~ seven property papers appear online last month, so at least there is some reading ahead of us! A mixture of privacy, quantum, high-dimensional distributions, and juntas (juntæ?). A lot of distribution testing, overall.

**Connecting Robust Shuffle Privacy and Pan-Privacy**, by Victor Balcer, Albert Cheu, Matthew Joseph, and Jieming Mao (arXiv). This paper considers a recent notion of differential privacy called* shuffle privacy*, where users have sensitive data, a central untrusted server wants to do something with that data (for instance, say… testing its distribution), and a trusted middle-man/entity shuffles the users’ messages u.a.r. to bring in a bit more anonymity. As it turns out, testing uniformity (or identity) of distributions in the shuffle privacy model is (i) much harder than without privacy constraints; (ii) much harder than with ‘usual’ (weaker) differential privacy (iii) much easier than with local privacy; (iv) related to the sample complexity under another privacy notion, *pan-privacy*. It’s a brand exciting new world out there!

*(Note: for the reader interested in keeping track of identity/uniformity testing of probability distributions under various privacy models, I wrote a very short summary of the current results here.)*

**Entanglement is Necessary for Optimal Quantum Property Testing, **by Sebastien Bubeck, Sitan Chen, and Jerry Li (arXiv). The analogue of uniformity testing, in the quantum world, is testing whether a quantum state is equal (or far from) the maximally mixed state. It’s known that this task has “quantum sample complexity” (number of measurements) \(\Theta(d/\varepsilon^2)\) (i.e., square root dependence on the dimension of the state, \(d^2\)). But this requires *entangled* measurements, which may be tricky to get (or, in my case, understand): what happens if the measurements can be adaptive, but not entangled? In this work, the authors show that, under this weaker access model \(\Omega(d^{4/3}/\varepsilon^2)\) measurements are necessary: adaptivity alone won’t cut it. It may still help though: without either entanglement *nor* adaptivity, the authors also show a \(\Omega(d^{3/2}/\varepsilon^2)\) measurements lower bound.

**Testing Data Binnings**, by Clément Canonne and Karl Wimmer (ECCC). More identity testing! Not private and not quantum for this one, but… not *quite* identity testing either. To paraphrase the abstract: this paper introduces (and gives near matching bounds for) the related question of *identity up to binning*, where the reference distribution \(q\) is over \(k \ll n\) elements: the question is then whether there exists a suitable binning of the domain \([n]\) into \(k\) intervals such that, *once binned*, \(p\) is equal to \(q\).”

**Hardness of Identity Testing for Restricted Boltzmann Machines and Potts models**, by Antonio Blanca, Zongchen Chen, Daniel Štefankovič, and Eric Vigoda (arXiv). Back to identity testing of distributions, but for high-dimensional structured ones this one. Specifically, this paper focuses on the undirected graphical models known as *restricted Boltzmann machines, *and provides efficient algorithms for identity testing and conditional hardness lower bounds depending on the type of correlations allowed in the graphical models.

**Robust testing of low-dimensional functions**, by Anindya De, Elchanan Mossel, and Joe Neeman (arXiv). Junta testing is a classical, central problem in property testing, with motivations and applications in machine learning and complexity. The related (and equally well-motivated) question of junta testing of functions on \(\mathbb{R}^d\) (instead of the Boolean hypercube) was recently studied by the same authors; and the related (and, again, equally well-motivated) question of *tolerant* junta testing on the Boolean hypercube was also recently studied (among other works) by the same authors. Well, this paper does it all, and tackles the challenging (and, for a change, equally well-motivated!) question of *tolerant* testing of juntas on \(\mathbb{R}^d\).

**Differentially Private Assouad, Fano, and Le Cam**, by Jayadev Acharya, Ziteng Sun, and Huanyu Zhang (arXiv). Back to probability distributions and privacy. This paper provides differentially private analogues of the classical eponymous statistical inference results (Assouad’s lemma, Fano’s inequality, and Le Cam’s method). In particular, it gives ready-to-use, blackbox tools to prove testing and learning lower bounds for distributions in the differentially private setting, and shows how to use them to easily derive, and rederive, several lower bounds.

**Edit: **We missed one!

**Learning and Testing Junta Distributions with Subcube Conditioning**, by Xi Chen, Rajesh Jayaram, Amit Levi, Erik Waingarten (arXiv). This paper focuses on the *subcube conditioning* model of (high-dimensional) distribution testing, where the algorithm can fix some variables to values of its choosing and get samples conditioned on those variables. Extending and refining techniques from a previous work by a (sub+super)set of the authors, the paper shows how to optimally learn and test junta distributions in this framework—with exponential savings with respect to the usual i.i.d. sampling model.