On the Wasserstein alignment problem
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arXiv
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| Format: | Preprint |
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2025
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| author | Pal, Soumik Sen, Bodhisattva Wong, Ting-Kam Leonard |
| author_facet | Pal, Soumik Sen, Bodhisattva Wong, Ting-Kam Leonard |
| contents | Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment problem seeks the transformation that minimizes the optimal transport cost between its pushforward of the source distribution and the target distribution, ensuring the closest possible alignment in a probabilistic sense. Examples of interest include two distributions on two Euclidean spaces $\mathbb{R}^n$ and $\mathbb{R}^d$, and we want a spatial embedding of the $n$-dimensional source measure in $\mathbb{R}^d$ that is closest in some Wasserstein metric to the target distribution on $\mathbb{R}^d$. Similar data alignment problems also commonly arise in shape analysis and computer vision. In this paper we show that this nonconvex optimal transport projection problem admits a convex Kantorovich-type dual. This allows us to characterize the set of projections and devise a linear programming algorithm. For certain special examples, such as orthogonal transformations on Euclidean spaces of unequal dimensions and the $2$-Wasserstein cost, we characterize the covariance of the optimal projections. Our results also cover the generalization when we penalize each transformation by a function. An example is the inner product Gromov-Wasserstein distance minimization problem which has recently gained popularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Wasserstein alignment problem Pal, Soumik Sen, Bodhisattva Wong, Ting-Kam Leonard Probability Optimization and Control 49Q22, 90C46 (Primary), 90C25, 90C05 (Secondary) Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment problem seeks the transformation that minimizes the optimal transport cost between its pushforward of the source distribution and the target distribution, ensuring the closest possible alignment in a probabilistic sense. Examples of interest include two distributions on two Euclidean spaces $\mathbb{R}^n$ and $\mathbb{R}^d$, and we want a spatial embedding of the $n$-dimensional source measure in $\mathbb{R}^d$ that is closest in some Wasserstein metric to the target distribution on $\mathbb{R}^d$. Similar data alignment problems also commonly arise in shape analysis and computer vision. In this paper we show that this nonconvex optimal transport projection problem admits a convex Kantorovich-type dual. This allows us to characterize the set of projections and devise a linear programming algorithm. For certain special examples, such as orthogonal transformations on Euclidean spaces of unequal dimensions and the $2$-Wasserstein cost, we characterize the covariance of the optimal projections. Our results also cover the generalization when we penalize each transformation by a function. An example is the inner product Gromov-Wasserstein distance minimization problem which has recently gained popularity. |
| title | On the Wasserstein alignment problem |
| topic | Probability Optimization and Control 49Q22, 90C46 (Primary), 90C25, 90C05 (Secondary) |
| url | https://arxiv.org/abs/2503.06838 |