December was indeed a merry month for property testing, with seven new papers appearing online.* Let’s hope the trend continues!

**Cube vs. Cube Low Degree Test**, by Amey Bhangale, Irit Dinur, and Inbal Livni Navon (ECCC). This work provides a new and improved analysis of the “low-degree test” of Raz and Safra, tightening the dependence on the alphabet size of the soundness parameter. Specifically, the goal is as follows: given query access to the encoding of a function \(f\colon \mathbb{F}^m\to \mathbb{F}\) as a “cube table,” decide whether \(f\) is a low-degree polynomial (i.e., of degree at most \(d\)). With direct applications to PCP theorems, this question has a long history; here, the authors focus on a very simple and natural test, introduced by Raz–Safra and Arora–Sudan. In particular, they improve the soundness guarantee, which previously required the error to be at least \(\textrm{poly}(d)/\mathbb{F}^{1/8}\), to obtain a dependence on the field size which is only \(\mathbb{F}^{-1/2}\).

**Robust Multiplication-based Tests for Reed-Muller Codes**, by Prahladh Harsha and Srikanth Srinivasan (arXiv). Given a function \(f\colon \mathbb{F}_q^n\to\mathbb{F}_q\) purported to be a degree-\(d\) polynomial, deciding whether this is indeed the case is a question with applications to hardness of approximation and — as the title strongly hints— to testing of Reed—Muller codes. Here, the authors generalize and improve on a test originally due to Dinur and Guruswami, improving on the soundness: that is, they show a robust version of the soundness guarantee of this multiplication test, answering a question left open by Dinur and Guruswami.

**A Note on Testing Intersection of Convex Sets in Sublinear Time**, by Israela Solomon (arXiv). In this note, the author addresses the following testing question: given \(n\) convex sets in \(\mathbb{R}^d\), distinguish between the case where (i) the intersection is non-empty, and (ii) even after removing any \(\varepsilon n\) sets, the intersection is still empty. Through the use of a generalization of Helly’s Theorem due Katchalski and Liu (1979), the author then provides and analyzes an algorithm for this question, with query complexity \(O(\log 1/(d\varepsilon))\).

**A Characterization of Constant-Sample Testable Properties**, by Eric Blais and Yuichi Yoshida (arXiv). A lot of work in property testing has been to understand which properties can be *locally* tested, i.e. admit testers with constant query complexity. Here, the authors tackle — and answer — the variant of this question in the *sample-based* testing model, that is restricting the ability of the algorithm to only obtain the value of the function on independently and uniformly distributed locations. Namely, they provide a full characterization of the properties of Boolean-valued function which are testable with constant sample complexity, resolving the natural question of whether most properties are easily tested from samples only. As it turns out, only those properties which are (essentially) \(O(1)\)-part-symmetric have this nice testability — where a property \(\mathcal{P}\) is \(k\)-part-symmetric if one can divide the input variables in \(k\) blocks, such that membership to \(\mathcal{P}\) is invariant by permutations inside each block.

The last three are all works on distribution testing, and all concerned with the *high-dimensional* case. Indeed, most of the distribution testing literature so far has been focusing in probability distributions over arbitrary discrete sets or the one-dimension ordered line; however, when trying to use these results in the (arbitrary) high-dimensional case on is subjected to the curse of dimensionality. *Can one leverage (presumed) structure to reduce the cost,* and obtain computationally and sample-efficient testers?

**Testing Ising Models**, by Costis Daskalakis, Nishanth Dikkala, and Gautam Kamath (arXiv). In this paper, the authors take on the above question in the context of Markov Random Fields, and more specifically in the Ising model. Modeling the (unknown) high-dimensional distributions as Ising models with some promise on the parameters, they tackle the two questions of *identity* and *independence* testing; respectively, “is the unknown Ising model equal to a fixed, known reference distribution?” and “is the high-dimensional, a priori complex distribution a product distribution?”. They obtain both testing algorithms and lower bounds for these two problems, where the soundness guarantee sought is in terms of the symmetrized Kullback—Leibler divergence (a notion of distance more stringent that the statistical distance).

**Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing**, by Costis Daskalakis and Qinxuan Pan (arXiv). Another natural model to study structured high-dimensional distribution is that of *Bayesian networks*, that is directed graphs providing a succinct description of the dependencies between coordinates. This paper studies the closeness testing problem (testing if two unknown distributions are equal or far, with regard to the usual statistical distance) for Bayesian networks, parameterized by the dimension, alphabet size, and (promise on the) maximum in-degree of the unknown Bayes net. At its core is a new inequality relating the (squared) Hellinger distance of two Bayesian networks to the sum of the (squared) Hellinger distances between their marginals.

**Testing Bayesian Networks**, by Clément Canonne, Ilias Diakonikolas, Daniel Kane, and Alistair Stewart (arXiv). Tackling the testing of Bayesian networks as well, this second paper considers two of the most standard testing questions — identity (one-unknown) and closeness (two-unknown) testing — under various assumptions, in order to pinpoint the *exact* sample complexity in each case. Specifically the goal is to see when (and under which natural restrictions) does testing become easier than learning for Bayesian networks, focusing on the dimension and maximum in-degree as parameters.

* As usual, if we forgot one or you find imprecisions in our review of a paper, please let us now in the comments below.