On Computing Total Variation Distance Between Mixtures of Product Distributions

• URL: http://arxiv.org/abs/2605.03839v1
• Date: Tue, 05 May 2026 15:05:49 GMT
• Title: On Computing Total Variation Distance Between Mixtures of Product Distributions
• Abstract summary: We study the problem of approximating the total variation distance between two mixtures of product distributions over a $n$-dimensional discrete domain.<n>For this class, we give a deterministic algorithm that exactly compute the total variation distance in time.
• Score: 7.048905632889667
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the problem of approximating the total variation distance between two mixtures of product distributions over an $n$-dimensional discrete domain. Given two mixtures $\mathbb{P}$ and $\mathbb{Q}$ with $k_1$ and $k_2$ product distributions over $[q]^n$, respectively, we give a randomized algorithm that approximates $d_{\mathrm{TV}}\left({\mathbb{P}},{\mathbb{Q}}\right)$ within a multiplicative error of $(1\pm \varepsilon)$ in time $\mathrm{poly}((nq)^{k_1+k_2},1/\varepsilon)$. We also study the special case of mixtures of Boolean subcubes over $\{0,1\}^n$. For this class, we give a deterministic algorithm that exactly computes the total variation distance in time $\mathrm{poly}(n,2^{O(k_1+k_2)})$, and show that exact computation is $\#\mathsf{P}$-hard when $k_1+k_2=Θ(n)$.

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