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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2406.07972 |
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| _version_ | 1866912150302228480 |
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| author | Erickson, William Q. |
| author_facet | Erickson, William Q. |
| contents | 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 $d$) 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 $n$-simplex. This recursion, however, was computationally expensive, requiring $\binom{d+n}{d}$ 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 $dn$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07972 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expected value and a Cayley-Menger type formula for the generalized earth mover's distance Erickson, William Q. Statistics Theory 60B05 (Primary) 49Q22 (Secondary) 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 $d$) 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 $n$-simplex. This recursion, however, was computationally expensive, requiring $\binom{d+n}{d}$ 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 $dn$. 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. |
| title | Expected value and a Cayley-Menger type formula for the generalized earth mover's distance |
| topic | Statistics Theory 60B05 (Primary) 49Q22 (Secondary) |
| url | https://arxiv.org/abs/2406.07972 |