Estimating Statistical Similarity Between Product Distributions
Keywords: total variation distance, TV distance, complement of total variation distance, TV similarity, statistical similarity, aggregated experts, FPTAS, reductions, Turing reductions, Knapsack, counting Knapsack, Masked Knapsack, counting Masked Knapsack
Abstract: We investigate the problem of computing the *statistical* or *total variation (TV) similarity* between distributions $P$ and $Q$, which is defined as $s_{\mathrm{TV}}(P,Q) := 1 - d_{\mathrm{TV}}(P, Q)$, where $d_{\mathrm{TV}}$ is the total variation distance between $P$ and $Q$.
Statistical similarity is a basic measure of similarity between distributions with several natural interpretations.
We focus on the case when $P$ and $Q$ are products of Bernoulli trials.
Recent work has established, somewhat surprisingly, that even for this simple class of distributions exactly computing the TV distance (and hence statistical similarity) is \#$\mathsf{P}$-hard.
This motivates the question of designing multiplicative approximation algorithms for these computational tasks.
It is known that the TV distance computation admits a fully polynomial-time deterministic approximation scheme (FPTAS).
It remained an open question whether efficient approximation schemes exist for estimating the statistical similarity between two product distributions.
In this work, we affirmatively answer this question by designing an FPTAS for estimating the statistical similarity between two product distributions.
To obtain our result, we introduce a new variant of the knapsack problem, which we call multidimensional Masked Knapsack problem, and design an FPTAS to estimate the number of solutions to this problem.
This result might be of independent interest.
Primary Area: learning theory
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Submission Number: 782
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