637,654 research outputs found

    Almost Optimal Distribution-Free Junta Testing

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    We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}^n. Chen, Liu, Servedio, Sheng and Xie [Zhengyang Liu et al., 2018] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O~(k^2)/epsilon queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O~(k/epsilon) queries

    Distribution-Free Proofs of Proximity

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    Motivated by the fact that input distributions are often unknown in advance, distribution-free property testing considers a setting in which the algorithmic task is to accept functions f:[n]{0,1}f : [n] \to \{0,1\} having a certain property Π\Pi and reject functions that are ϵ\epsilon-far from Π\Pi, where the distance is measured according to an arbitrary and unknown input distribution D[n]D \sim [n]. As usual in property testing, the tester is required to do so while making only a sublinear number of input queries, but as the distribution is unknown, we also allow a sublinear number of samples from the distribution DD. In this work we initiate the study of distribution-free interactive proofs of proximity (df-IPP) in which the distribution-free testing algorithm is assisted by an all powerful but untrusted prover. Our main result is a df-IPP for any problem ΠNC\Pi \in NC, with O~(n)\tilde{O}(\sqrt{n}) communication, sample, query, and verification complexities, for any proximity parameter ϵ>1/n\epsilon>1/\sqrt{n}. For such proximity parameters, this result matches the parameters of the best-known general purpose IPPs in the standard uniform setting, and is optimal under reasonable cryptographic assumptions. For general values of the proximity parameter ϵ\epsilon, our distribution-free IPP has optimal query complexity O(1/ϵ)O(1/\epsilon) but the communication complexity is O~(ϵn+1/ϵ)\tilde{O}(\epsilon \cdot n + 1/\epsilon), which is worse than what is known for uniform IPPs when ϵ<1/n\epsilon<1/\sqrt{n}. With the aim of improving on this gap, we further show that for IPPs over specialised, but large distribution families, such as sufficiently smooth distributions and product distributions, the communication complexity can be reduced to ϵn(1/ϵ)o(1)\epsilon\cdot n\cdot(1/\epsilon)^{o(1)} (keeping the query complexity roughly the same as before) to match the communication complexity of the uniform case

    On sample-based testers

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    The standard definition of property testing endows the tester with the ability to make arbitrary queries to “elements ” of the tested object. In contrast, sample-based testers only obtain independently distributed elements (a.k.a. labeled samples) of the tested object. While samplebased testers were defined by Goldreich, Goldwasser, and Ron (JACM 1998), most research in property testing is focused on query-based testers. In this work, we advance the study of sample-based property testers by providing several general positive results as well as by revealing relations between variants of this testing model. In particular: • We show that certain types of query-based testers yield sample-based testers of sublinear sample complexity. For example, this holds for a natural class of proximity oblivious testers. • We study the relation between distribution-free sample-based testers and one-sided error sample-based testers w.r.t the uniform distribution. While most of this work ignores the time complexity of testing, one part of it does focus on this aspect. The main result in this part is a sublinear-time sample-based tester for k-Colorability, for any k ≥ 2

    A Nonparametric Test For Homogeneity Of Variances: Application To GPAs Of Students Across Academic Majors

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    We propose a nonparametric (or distribution-free) procedure for testing the equality of several population variances (or scale parameters). The proposed test is a modification of Bakir&rsquo;s (1989, Commun. Statist., Simul-Comp., 18, 757-775) analysis of means by ranks (ANOMR) procedure for testing the equality of several population means. A proof is given to establish the distribution-free property of the modified procedure. The proposed procedure is then applied to test whether or not the variability in the grade point averages (GPAs) of students differs across five business academic majors. We collect the GPAs (observations) of a random sample of students from each major under study. The absolute deviations of the observations from the overall median of the combined sample are then calculated and ranked from least to largest. The average ranks and two decision lines are then plotted on a graph paper to detect not only the existence of significant differences among variances, but also to pinpoint which variances are causing those differences
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