On the Wasserstein alignment problem

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.