On Closeness to k-Wise Uniformity

A probability distribution over {-1, 1}^n is (epsilon, k)-wise uniform if, roughly, it is epsilon-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (epsilon, k)-wise uniform distribution can be from any globally k-wise uniform distribution. We show that every (epsilon, k)-wise uniform distribution is O(n^{k/2}epsilon)-close to a k-wise uniform distribution in total variation distance. In addition, we show that this bound is optimal for all even k: we find an (epsilon, k)-wise uniform distribution that is Omega(n^{k/2}epsilon)-far from any k-wise uniform distribution in total variation distance. For k=1, we get a better upper bound of O(epsilon), which is also optimal. One application of our closeness result is to the sample complexity of testing whether a distribution is k-wise uniform or delta-far from k-wise uniform. We give an upper bound of O(n^{k}/delta^2) (or O(log n/delta^2) when k = 1) on the required samples. We show an improved upper bound of O~(n^{k/2}/delta^2) for the special case of testing fully uniform vs. delta-far from k-wise uniform. Finally, we complement this with a matching lower bound of Omega(n/delta^2) when k = 2. Our results improve upon the best known bounds from [Alon et al., 2007], and have simpler proofs

Finding Skewed Subcubes Under a Distribution

Say that we are given samples from a distribution ? over an n-dimensional space. We expect or desire ? to behave like a product distribution (or a k-wise independent distribution over its marginals for small k). We propose the problem of enumerating/list-decoding all large subcubes where the distribution ? deviates markedly from what we expect; we refer to such subcubes as skewed subcubes. Skewed subcubes are certificates of dependencies between small subsets of variables in ?. We motivate this problem by showing that it arises naturally in the context of algorithmic fairness and anomaly detection. In this work we focus on the special but important case where the space is the Boolean hypercube, and the expected marginals are uniform. We show that the obvious definition of skewed subcubes can lead to intractable list sizes, and propose a better definition of a minimal skewed subcube, which are subcubes whose skew cannot be attributed to a larger subcube that contains it. Our main technical contribution is a list-size bound for this definition and an algorithm to efficiently find all such subcubes. Both the bound and the algorithm rely on Fourier-analytic techniques, especially the powerful hypercontractive inequality. On the lower bounds side, we show that finding skewed subcubes is as hard as the sparse noisy parity problem, and hence our algorithms cannot be improved on substantially without a breakthrough on this problem which is believed to be intractable. Motivated by this, we study alternate models allowing query access to ? where finding skewed subcubes might be easier

Succinct quantum testers for closeness and kk-wise uniformity of probability distributions

We explore potential quantum speedups for the fundamental problem of testing the properties of closeness and $k$-wise uniformity of probability distributions. \textit{Closeness testing} is the problem of distinguishing whether two $n$-dimensional distributions are identical or at least $\varepsilon$-far in $\ell^1$- or $\ell^2$-distance. We show that the quantum query complexities for $\ell^1$- and $\ell^2$-closeness testing are O\rbra{\sqrt{n}/\varepsilon} and O\rbra{1/\varepsilon}, respectively, both of which achieve optimal dependence on $\varepsilon$, improving the prior best results of \hyperlink{cite.gilyen2019distributional}{Gily{\'e}n and Li~(2019)}. \textit{$k$-wise uniformity testing} is the problem of distinguishing whether a distribution over \cbra{0, 1}^n is uniform when restricted to any $k$ coordinates or $\varepsilon$-far from any such distributions. We propose the first quantum algorithm for this problem with query complexity O\rbra{\sqrt{n^k}/\varepsilon}, achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity O\rbra{n^k/\varepsilon^2} by \hyperlink{cite.o2018closeness}{O'Donnell and Zhao (2018)}. Moreover, when $k = 2$ our quantum algorithm outperforms any classical one because of the classical lower bound \Omega\rbra{n/\varepsilon^2}. All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.Comment: We have added the proof of lower bounds and have polished the languag