2025/03/10 by Pal, Soumik, Sen, Bodhisattva, Wong, Ting-Kam Leonard · 2 citations
#49Q22 #90C05 (Secondary) #90C25 #90C46 (Primary) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)
paper · doi:10.48550/arxiv.2503.06838
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 ℝn and ℝd, and we want a spatial embedding of the n-dimensional source measure in ℝd that is closest in some Wasserstein metric to the target distribution on ℝd. 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.