Happy Fourth of July to everyone! Last month was prolific for property testing, as we counted no less than nine papers making their way online!*

**Parameterized Property Testing of Functions**, by Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma. (ECCC) *(A preliminary version was published in ITCS’17)*. In property testing, here of functions, it is standard to parameterize the query complexity by the two “obvious” parameters — namely, the domain size parameter \(n\), and the proximity parameter \(\varepsilon\). While this often leads to a good understanding of the landscape, sometime a more fine-grained analysis may be useful, to capture the complexity of the question in terms of the setting-specific “right” parameters. This work initiates this line of inquiry for functions \(f\colon [n] \to \mathbb{R}\), showing examples where classic lower bounds can be circumvented by focusing on a more relevant parameters of the problem.

**A Lower Bound for Nonadaptive, One-Sided Error Testing of Unateness of Boolean Functions over the Hypercube**, by Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, C. Seshadhri (arXiv). A Boolean function \(f\colon \{0,1\}^n\to \{0,1\}\) is said to be *unate* if it is monotone (either increasing or decreasing) in each coordinate; or, equivalently, if there exists \(\sigma\in\{0,1\}^n\) such that \(x\mapsto f(x\oplus\sigma)\) is monotone. As reported in the previous months, there has been a significant amount of work lately on testing unateness of functions (including real-valued). In this short paper, the authors improve a lower bound of Chen et al. (2017) on *non-adaptive, one-sided* unateness testing of Boolean functions, bringing it from \(\Omega(\frac{n}{\log^2 n})\) to \(\Omega(\frac{n}{\log n})\) — leaving only a gap of \(\log^2 n\) between known upper and lower bounds.

**Adaptivity is exponentially powerful for testing monotonicity of halfspaces**, by Xi Chen, Rocco A. Servedio, Li-Yang Tan, and Erik Waingarten (arXiv:). While monotonicity testing of general Boolean functions has been widely studied these past years, testing monotonicity of functions promised to belong to a specific class (here, halfspaces — linear threshold functions) is much less understood. In this work, the authors show that testing monotonicity of halfspaces can done adaptively with \(\mathrm{poly}(\log n, 1/\varepsilon)\) queries. Since the \(\Omega(n^{1/2})\) non-adaptive lower bound for general monotonicity testing relied on instances that were halfspaces, it applies here as well — showing an exponential gap between adaptive and non-adaptive testing in this case. *“It’s as extreme as it gets!”*

**Testing Piecewise Functions**, by Steve Hanneke and Liu Yang (arXiv). Generalizing the concept of “unions of intervals,” the authors consider here the following general question: fix a class of functions \(\mathcal{H}\subseteq \mathcal{Y}^{\mathcal{X}}\), where \(\mathcal{X},\mathcal{Y}\) are “nice” spaces (typically \(\mathcal{X}=\mathbb{R}\)) and parameter \(k\) and \(\varepsilon\). The goal is to test whether (i) one can partition \(\mathcal{X}\) into \(k\) pieces and find \(h_1,\dots,h_k\in\mathcal{H}\), such that \(f=h_i\) on the \(i\)-th piece; or (ii) \(f\) is \(\varepsilon\)-far (for a notion of distance depending on the measure on \(\mathcal{X}\)) from every such “\(k\)-piecewise function.”

They give upper bounds (as well as a lower bound) for the query complexity of this question in both the active testing and the sample-based testing settings (for the latter, considering the class \(\mathcal{H}\) of constant functions).

**Sample-based high-dimensional convexity testing**, by Xi Chen, Adam Freilich, Rocco A. Servedio, and Timothy Sun (arXiv). In the setting considered in this paper, there exists an unknown set \(S\subseteq \mathbb{R}^n\), and the algorithm is provided with *samples* \(\langle x, S(x)\rangle\), where \(x\) is distributed according to a standard Gaussian distribution and the label \(S(x)\) is \(1\) iff \(x\in S\). The goal is to test whether \(S\) is convex (or $\varepsilon$-far from convex under the standard Gaussian distribution). The authors then show near-matching upper and lower bounds: one-sided testing of convexity in this testing has sample complexity \(2^{\tilde{\Theta}(n)}\), while two-sided testing has sample complexity \(2^{\tilde{\Theta}(\sqrt{n})}\).

**Distributed Detection of Cycles**, by Pierre Fraigniaud and Dennis Olivetti (arXiv); and **Distributed Subgraph Detection**, by Pierre Fraigniaud, Pedro Montealegre, Dennis Olivetti, Ivan Rapaport, and Ioan Todinca (arXiv). Improving the understanding of distributed property testing of graphs in the CONGEST model (already touched upon last month), these two works tackle the question of distributed testing of subgraph freeness. The first considers testing cycle-freeness, showing that \(C_k\)-freeness can be tested in the CONGEST model in a constant number of rounds (and with one-sided error). The second considers the more general question testing \(H\)-freeness of graphs, for a larger class of subgraphs that includes cycles. Specifically, they show that for any pattern \(H\) consisting of a couple of nodes, connected in an arbitrary manner to a tree, \(H\)-freeness can be tested (with one-sided error) in \(O(1)\) rounds.

**Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates**, by Sivaraman Balakrishnan and Larry Wasserman (arXiv). In distribution testing, the question of testing identity to a known discrete distribution \(p\) is almost fully settled, with near tight “instance-optimal” bounds. But what about the related question of testing identity to \(p\) (not necessarily discrete) under the additional promise that both \(p\) and the unknown distribution \(q\) both belong to a specific class \(C\) of distributions, instead of arbitrary? This is the problem this work tackles, for the specific case of \(C\) being (i) the class of multinomial distributions, and (ii) the class of (continuous) Lipschitz distributions. In both cases, the authors establish some near-tight bounds, both of an “instance-optimal” flavor.

**On Axis-Parallel Tests for Tensor Product Codes**, by Alessandro Chiesa, Peter Manohar, and Igor Shinkar (ECCC). Low-degree testing has its roots in the origins of property testing, and its connection to the PCP theorem; and has been a fruitful and rich line of work since. Here, the authors analyze a type of low-degree test introduced in 2006 by Ben-Sasson and Sudan, which constrains the tester to only considers restrictions along axis-parallel hyperplanes; and establish new results on these class of (weaker but more general) testers.

** As usual, if you find a paper we forgot or misrepresented, please signal it in the comments below.*