On the Wasserstein alignment problem

TL;DR

This work introduces the Wasserstein downward alignment problem, where a family of transformations $\{T_{\theta}\}$ maps a source measure $\mu$ in $\mathcal{X}$ to a target space $\mathcal{Z}$ with measure $\nu$, and seeks the transformation minimizing the Wasserstein cost between $(T_{\theta})_{\#}\mu$ and $\nu$ via $\mathbb{W}_c^{\downarrow}(\mu,\nu)$. A key contribution is a generalized Kantorovich duality that yields a convex relaxation and a tractable dual formulation, enabling a linear-programming algorithm for empirical/discrete data and extensions to penalized objectives $\mathbb{W}_c^{\downarrow,R}(\mu,\nu)$. In the Euclidean setting, the paper shows an equivalence between downward and upward formulations under zero-mean, unit-covariance normalization, and derives a first-order optimality condition for orthogonal projections, including a concrete 2D example highlighting nonconvexity and local minima. The implementation sections provide a concrete LP approach for empirical measures and demonstrate alignment of 2D point clouds, illustrating the method’s robustness and ability to preserve geometric structure. Overall, the results offer a principled, convex-analytic framework for cross-space distribution alignment with practical LP-based computability and broad applicability to shape analysis and related tasks.

Abstract

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.

Figures (4)

• Figure 1: Illustration of the Wasserstein alignment problem \ref{['eq:wgencost']}, where both $\mu$ and $\nu$ are discrete measures. Given a family $\{T_{\theta}\}_{\theta \in \Theta}$ of continuous mappings from $\mathcal{X}$ to $\mathcal{Z}$, we wish to find $\theta_* \in \Theta$ such that $(T_{\theta_*})_{\#} \mu$ is closest to $\nu$ with respect to the optimal transport cost $\mathbb{W}_c$. Left: $\mu$ is the point cloud with points $x_i$ represented by solid red circles. Right: $\nu$ is the point cloud with points $z_j$ represented by hollow blue circles. The solid curve and the solid red circles represent the images of $\mathcal{X}$ and $x_i$ under the optimal map $T_{\theta_*}$. The other curves and circles are images under suboptimal $T_{\theta}$.
• Figure 2: Empirical illustration of the Euclidean case and Proposition \ref{['prop:Euclidean.equivalence']}.
• Figure 3: Graphs of the function $F(t)$ from \ref{['eq:whatisF']} as $c$ increases from $0$ (light grey) to $\infty$ (black). The thick blue curve corresponds to $c = 1$.
• Figure 4: Left: The discrete distribution $\mu$ with $N = 150$ equally weighted atoms. Center: The discrete distribution $\nu$ with $M= 80$ equally weighted atoms. Right: The two distributions optimally aligned.

Theorems & Definitions (25)

• Definition 1: Function class for the dual problem
• Theorem 1: Generalized Kantorovich duality
• Definition 2: $c/\overline{c}$- transform
• Corollary 1: Optimality criterion
• Lemma 1
• proof
• Proposition 1: Existence and stability
• proof
• Lemma 2
• proof