A simple polynomial-time approximation algorithm for the total variation distance between two product distributions

Feng, Wei; Guo, Heng; Jerrum, Mark; Wang, Jiaheng

doi:10.1137/1.9781611977585.ch30

Cited by 1 publication

(4 citation statements)

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“…Prior to our work, such approximation schemes were known only for product distributions, which are Bayes nets over a graph with no edges [FGJW23]. In particular, designing an FPRAS for estimating TV distance between Bayes nets over trees (which are graphs with treewidth 1) was an open question.…”

Section: Corollary 3 (Informal)mentioning

confidence: 99%

“…In that work, they proved that exactly computing the TV distance between product distributions is #P-complete, that it is NP-hard to decide whether the TV distance between two Bayes nets of in-degree 2 is equal to 0 or not, and also gave an FPTAS for approximating the TV distance between an arbitrary product distribution and the uniform distribution. In a subsequent work, Feng, Guo, Jerrum and Wang [FGJW23] gave an FPRAS for approximating the TV distance between two arbitrary product distributions.…”

Section: Distance Computationmentioning

confidence: 99%

“…In a very recent work, Feng, Guo, Jerrum, and Wang [FGJW23] showed that, when P and Q are product distributions, approximating Pr L [X = Y ] nevertheless always leads to a good approximation for Pr O [X = Y ] where L and O are local and optimal couplings respectively of P and Q. More precisely, denoting by α the ratio [FGJW23] showed are that for any two n-dimensional product distributions P and Q:…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Our proof of this inequality is elementary but crucially uses properties (ii) and (iii) in the definition of L above. Now we can follow the same strategy as [FGJW23] by defining an unbiased estimator α for α = d TV (P, Q)/Z and empirically estimating the expectation of α.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 3 more Smart Citations

On Approximating Total Variation Distance

Bhattacharyya

Gayen

Meel

et al. 2023

Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence

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Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the problem of computing the TV distance of two product distributions over the domain {0,1}^n. In particular, we establish the following results. 1. The problem of exactly computing the TV distance of two product distributions is #P-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals leading to efficient algorithms. 2. There is a fully polynomial-time deterministic approximation scheme (FPTAS) for computing the TV distance of two product distributions P and Q where Q is the uniform distribution. This result is extended to the case where Q has a constant number of distinct marginals. In contrast, we show that when P and Q are Bayes net distributions the relative approximation of their TV distance is NP-hard.

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Section: Corollary 3 (Informal)mentioning

confidence: 99%

Section: Distance Computationmentioning

confidence: 99%