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1. Sharp Constants in Uniformity Testing via the Huber Statistic

   Gupta, S ; Price, Eric ( January 2022 , Conference on Learning Theory)

   Uniformity testing is one of the most well-studied problems in property testing, with many known test statistics, including ones based on counting collisions, singletons, and the empirical TV distance. It is known that the optimal sample complexity to distinguish the uniform distribution on m elements from any ϵ-far distribution with 1−δ probability is n=Θ(mlog(1/δ)√ϵ2+log(1/δ)ϵ2), which is achieved by the empirical TV tester. Yet in simulation, these theoretical analyses are misleading: in many cases, they do not correctly rank order the performance of existing testers, even in an asymptotic regime of all parameters tending to 0 or ∞. We explain this discrepancy by studying the \emph{constant factors} required by the algorithms. We show that the collisions tester achieves a sharp maximal constant in the number of standard deviations of separation between uniform and non-uniform inputs. We then introduce a new tester based on the Huber loss, and show that it not only matches this separation, but also has tails corresponding to a Gaussian with this separation. This leads to a sample complexity of (1+o(1))mlog(1/δ)√ϵ2 in the regime where this term is dominant, unlike all other existing testers. 

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2. Testing Mixtures of Discrete Distributions

   Aliakbarpour, M. ; Kumar, R. ; Rubinfeld, R ( January 2019 , Proceedings of the Thirty-Second Conference on Learning Theory (COLT 2019))

   There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size n, distinguishing the uniform distribution from distributions that are far from uniform in ℓ1-distance uses only O(n−−√) samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is ϵ-close to uniform from the case where the distribution is (1−ϵ)-far from uniform. The latter task requires nearly linear in n samples (Valiant, 2008; Valiant and Valiant, 2017a). In this work, we present a noise model that on one hand is more tractable for the testing problem, and on the other hand represents a rich class of noise families. In our model, the noisy distribution is a mixture of the original distribution and noise, where the latter is known to the tester either explicitly or via sample access; the form of the noise is also known \emph{a priori}. Focusing on the identity and closeness testing problems leads to the following mixture testing question: Given samples of distributions p,q1,q2, can we test if p is a mixture of q1 and q2? We consider this general question in various scenarios that differ in terms of how the tester can access the distributions, and show that indeed this problem is more tractable. Our results show that the sample complexity of our testers are exactly the same as for the classical non-mixture case. 

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3. Testing Mixtures of Discrete Distributions

   Aliakbarpour, Maryam ; Kumar, Ravi ; Rubinfeld, Ronitt ( January 2019 , Proceedings of the Thirty-Second Conference on Learning Theory (PMLR))

   There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size n, distinguishing the uniform distribution from distributions that are far from uniform in ℓ1-distance uses only O(n−−√) samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is ϵ-close to uniform from the case where the distribution is (1−ϵ)-far from uniform. The latter task requires nearly linear in n samples (Valiant, 2008; Valiant and Valiant, 2017a). In this work, we present a noise model that on one hand is more tractable for the testing problem, and on the other hand represents a rich class of noise families. In our model, the noisy distribution is a mixture of the original distribution and noise, where the latter is known to the tester either explicitly or via sample access; the form of the noise is also known \emph{a priori}. Focusing on the identity and closeness testing problems leads to the following mixture testing question: Given samples of distributions p,q1,q2, can we test if p is a mixture of q1 and q2? We consider this general question in various scenarios that differ in terms of how the tester can access the distributions, and show that indeed this problem is more tractable. Our results show that the sample complexity of our testers are exactly the same as for the classical non-mixture case. 

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4. A $d^{1/2+o(1)}$ Monotonicity Tester for Boolean Functions on $d$-Dimensional Hypergrids

   Hadley Black, Deeparnab Chakrabarty ( November 2023 , Annual Symposium on Foundations of Computer Science)

   Monotonicity testing of Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $n$, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity $\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$. This complexity is independent of $n$, but has a suboptimal dependence on $d$. Recently, [Braverman-Khot-Kindler-Minzer, ITCS 2023] and [Black-Chakrabarty-Seshadhri, STOC 2023] describe $\widetilde{O}(\varepsilon^{-2} n^3\sqrt{d})$ and $\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$-query testers, respectively. These testers have an almost optimal dependence on $d$, but a suboptimal polynomial dependence on $n$. \smallskip In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity $O(\varepsilon^{-2} d^{1/2 + o(1)})$, \emph{independent} of $n$. Up to the $d^{o(1)}$-factors, our result resolves the non-adaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of $n$ yields a non-adaptive, one-sided $O(\varepsilon^{-2} d^{1/2 + o(1)})$-query monotonicity tester for Boolean functions $f:\mathbb{R}^d \to \{0,1\}$ associated with an arbitrary product measure. 

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5. Sampling Multiple Edges Efficiently

   Eden, T. ; Mossel, S. ; Rubinfeld, R. ( January 2021 , Random)

   We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ϵ-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ∗(n/ √ m) for sampling a single edge in general graphs (where O ∗(·) suppresses poly(1/ϵ) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O∗(√ q · (n/ √ m)), which is strictly preferable to O∗(q · (n/ √ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal. 

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