Distance Estimation for High-Dimensional Distributions
Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
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Keywords: Sampling, Distribution Testing, High-dimensional statistics
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Abstract: We study the distance estimation problem for high-dimensional distributions. Given two distributions $\mathcal{P}$ and $\mathcal{Q}$ over $\{0,1\}^n$, and a parameter $\varepsilon$, the goal of distance estimation is to determine the statistical distance between the two distributions up to an additive tolerance $\pm \varepsilon$. Since exponential lower bounds (in $n$) are known for the problem in the standard sampling model, research has focused on models where one can draw conditional samples.
Among these models, \textit{subcube conditioning} ($\mathsf{SUBCOND}$), i.e., conditioning on arbitrary subcubes of the domain, holds the promise of widespread practical adoption owing to its ability to capture the natural behavior of distribution samplers. In this paper, we present the first polynomial sample distance estimator in the conditional sampling model, and our algorithm makes $\tilde{\mathcal{O}}(n^3/\varepsilon^5)$ \subcond queries. We implement our algorithm to estimate the distance between distributions arising from real-life sampling benchmarks, and we find that our algorithm easily scales beyond the naive method.
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Submission Number: 651
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