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arXiv:1911.07357 · Electronic colloquium on computational complexity

Random Restrictions of High-Dimensional Distributions and Uniformity Testing with Subcube Conditioning

Clément L. Canonne, Chen, Xi, Gautam Kamath, Amit Levi, Erik Waingarten

Published Nov 17, 2019• 0 citations
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Abstract

We give a nearly-optimal algorithm for testing uniformity of distributions supported on $\{-1,1\}^n$, which makes $\tilde O (\sqrt{n}/\varepsilon^2)$ queries to a subcube conditional sampling oracle (Bhattacharyya and Chakraborty (2018)). The key technical component is a natural notion of random restriction for distributions on $\{-1,1\}^n$, and a quantitative analysis of how such a restriction affects the mean vector of the distribution. Along the way, we consider the problem of mean testing with independent samples and provide a nearly-optimal algorithm.

DOI
10.48550/arxiv.1911.07357

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Property testing of distributions is a central topic in information theory, learning theory, and statistics. While quantum algorithms are known to offer significant speedups for property testing of a single distribution or a pair of distributions, it is unclear whether quantum algorithms provide speedups for property testing of mmm (m≥3m\geq 3m≥3) distributions. This work focuses on quantum algorithms for testing whether mmm distributions have similar means or are εεε-far from mean similarity under two models. In the query model, the algorithm can choose which distribution to sample from, whereas in the sampling model, the distributions are selected uniformly. We design quantum algorithms with complexities O~(1/ε)\tilde{O}(1/ε)O~(1/ε) (the O~\tilde{O}O~ notation hides poly-logarithmic factors) and O~(m/ε)\tilde{O}(\sqrt{m}/ε)O~(m​/ε) in the query and sampling models, respectively, achieving quadratic speedups over the classical counterparts. We further establish quantum lower bounds of Ω(1/ε)Ω\left(1/ε\right)Ω(1/ε) and $Ω\rbra{m^{1/3}+\frac{m^{1/4}}ε}$ for the query model and the sampling model, demonstrating the optimality of our quantum algorithms in terms of the dependence on εεε up to logarithmic factors.
We explore potential quantum speedups for the fundamental problem of testing the properties of closeness andk-wise uniformity of probability distributions. •Closeness testingis the problem of distinguishing whether twon-dimensional distributions are identical or at least ε-far in ℓ1- or ℓ2-distance. We show that the quantum query complexities for ℓ1- and ℓ2-closeness testing areO(√n/ε) andO(1/ε), respectively, both of which achieve optimal dependence on ε, improving the prior best results of Gilyén and Li (2019). •k-wise uniformity testingis the problem of distinguishing whether a distribution over {0, 1}nis uniform when restricted to anykcoordinates or ε-far from any such distribution. We propose the first quantum algorithm for this problem with query complexityO(√nk/ε), achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexityO(nk/ε2) by O’Donnell and Zhao (2018). Moreover, whenk= 2 our quantum algorithm outperforms any classical one because of the classical lower bound Ω(n/ε2). All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.
Suppose one has access to oracles generating samples from two unknown probability distributions p and q on some N-element set. How many samples does one need to test whether the two distributions are close or far from each other in the L1-norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L1-distance ∥p-q∥1can be estimated with a constant precision using only O(N1/2) queries in the quantum settings, whereas classical computers need Ω(N1-o(1)) queries. We also describe quantum algorithms for testing uniformity and orthogonality with query complexity O(N1/3). The classical query complexity of these problems is known to be Ω(N1/2).
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