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Abstract
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The earth mover’s distance (EMD), also known as the 1-Wasserstein metric, measures the
minimum amount of work required to transform one probability distribution into another.
The EMD can be naturally generalized to measure the “distance” between any number
(say
) of
distributions. In previous work (2021), we found a recursive formula for the expected
value of the generalized EMD, assuming the uniform distribution on the standard
-simplex.
This recursion, however, was computationally expensive, requiring
many
iterations. The main result of the present paper is a nonrecursive formula for this
expected value, expressed as the integral of a certain polynomial of degree at most
. As a
secondary result, we resolve an unanswered problem by giving a formula for the
generalized EMD in terms of pairwise EMDs; this can be viewed as an analogue of
the Cayley–Menger determinant formula that gives the hypervolume of a simplex in
terms of its edge lengths.
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Keywords
earth mover's distance, Wasserstein metric,
Cayley–Menger-type formulas
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Mathematical Subject Classification
Primary: 60B05
Secondary: 49Q22
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Milestones
Received: 12 June 2024
Revised: 6 October 2024
Accepted: 1 November 2024
Published: 10 December 2024
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