On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs

TL;DR

This work generalizes optimal transport to vector fields on connection graphs by formulating a Beckmann-type cost $W_1^{\sigma}(\alpha,\beta)$ that uses a parallel-transport-aware incidence $B$ and per-edge weights $w(e)$. It develops feasibility conditions tied to the connection Laplacian $L$, establishes strong duality for both the unregularized and a relaxed-regularized variant, and shows how switching equivalence can render natural vector-valued densities feasible. A practical dual-derivation yields a closed-form primal update, enabling scalable gradient-based methods, and the framework is demonstrated on color image transport, vector-field interpolation on manifolds, and hurricane trajectory clustering. Overall, the paper provides a theoretically solid, computationally tractable extension of Wasserstein-type transport to vector-valued data on graphs, with broad applicability to geometric data, physics-inspired flows, and environmental trajectory analysis.

Abstract

We propose a model of optimal parallel transport between vector fields on a connection graph, which consists of a weighted graph along with a map from its edges to an orthogonal group. Inspired by the well-known equivalence of 1-Wasserstein distance and minimum cost flows on standard graphs, we consider two versions of this problem: a minimum norm vector-valued flow problem with divergence constraints reflective of the connection structure of the graph; and a modified version which incorporates both quadratic regularization and a relaxation of the divergence constraint. Our theoretical contributions include: conditions for feasibility and computation of the Lagrangian dual problem for both problems, and duality correspondence for the relaxed-regularized version. Example applications of the model including transport between color images, vector field interpolation, and unsupervised clustering of vector field-valued data (in this case hurricane trajectory data) are also considered.

Figures (6)

• Figure 1: Here, $d=1$ and $\sigma$ is a $\pm 1$ connection on each edge. The non-zero elements of $\alpha$ and $\beta$ are displayed. (a) Case where $\alpha$ and $\beta$ have equal mass but no feasible flow. As $\alpha$ undergoes parallel transport in the direction of node $3$, its sign is flipped, which is therefore not compatible with $\beta$. Instead, $\beta(3) = -1$ would result in a feasible problem. (b) Case where $\alpha$ and $\beta$ do not have equal total mass but the problem is feasible. Take $J = 0.25$ on the upper path and $J = 0.75$ on the lower path, and it follows that $BJ = \alpha-\beta$ due to \ref{['eq:incidence-matrix']}.
• Figure 2: Visualizations of the experiment described in \ref{['ex:cat-horse']}. (a)-(b) Illustrations of the images which are used to construct $\alpha,\beta$ upon sampling to the nodes of the $48\times 48$ lattice graph $G$. (c)-(f) Visualizing the optimal flows $J_{\delta,\lambda}$ where $\delta$ is fixed and $\lambda=1,10,100,10^3$. We overlay $\alpha,\beta$ in the images to provide a spatial reference for the flows. The procedure by which we are rendering $J_{\delta,\lambda}$ is described in \ref{['ex:cat-horse']}. (g)-(j) Visualizing the optimal flows $J_{\delta,\lambda}$ where $\lambda=10^2$ is fixed and $\delta=\delta^\ast,2\delta^\ast, 3\delta^\ast, 4\delta^\ast$. Here, $\delta^\ast$ is as in \ref{['eq:delta-star']}.
• Figure 3: (Left) An illustration of the graph $G_t$ obtained by sampling points from the parameter domain $(\theta, \psi) \in [0, 2\pi]^2$ of the standard parameterization of a torus $(\theta, \psi)\mapsto ((R + r\cos\theta)\cos\psi, (R + r\cos\theta)\sin\psi, r\sin\theta)$ for $R=5$ and $r=1$ with $\epsilon = 1$. Following the procedure and notation outlined at the beginning of \ref{['subsec:local-pca']}, we obtain the orthonormal vectors $O_i$ approximating the tangent space at each node, and render them in orange and purple for 5% of the nodes selected uniformly at random. (Right) Similarly, an illustration of the graph $G_b$ obtained by sampling 1500 points from the Stanford Bunny 3D mesh turk1994zippered with $\epsilon = 0.015$ as well as, for 5% of the nodes, the orthonormal vectors $O_i$ obtained from the procedure outlined in \ref{['subsec:local-pca']} shown in orange and purple.
• Figure 4: Continuing from \ref{['fig:local-pca-drawing']}, we implement the vector field trajectory interpolation algorithm in \ref{['ex:vector-field-reconstruction']} using the graph $G_t$, with $O_i\alpha_k(i)$ shown at each node $i$ in red for step $k=0$ and $O_i\beta(i)$ at step twenty. $\alpha$ (resp. $\beta$) was chosen by setting $\alpha(i) = O_i^\dagger001^\top$ (resp. $\beta(i) = O_i^\dagger00-1^\top$) whenever $\|x_i - s \| < 0.75$ (resp. $\|x_i - t \| < 0.75$) and zero otherwise where $s\in\mathbb{R}^3$ (resp. $t\in\mathbb{R}^3$) is the right-most (resp. left-most) endpoint of $G_t$ and $O_i^\dagger$ is the matrix pseudo-inverse of $O_i$. $\lambda=100$ and $\delta\approx 10^{-7}$ were used for this experiment. Vector field colorings are obtained from a linear interpolation from red to blue independent of the vector fields or optimal flows.
• Figure 5: Continuing from \ref{['fig:local-pca-drawing']}, we implement the vector field trajectory interpolation algorithm in \ref{['ex:vector-field-reconstruction']} using the graph $G_b$, with $O_i\alpha_k(i)$ shown at each node $i$ in red for step $k=0$ and $O_i\beta(i)$ at step fifteen. $\alpha$ (resp. $\beta$) was chosen by setting $\alpha(i) = O_i^\dagger001^\top$ (resp. $\beta(i) = O_i^\dagger00-1^\top$) whenever $\|x_i - s \| < 0.05$ (resp. $\|x_i - t \| < 0.05$) and zero otherwise where $s, t\in\mathbb{R}^3$ are two points on the mesh chosen without particular preference and $O_i^\dagger$ is the matrix pseudo-inverse of $O_i$. $\lambda=100$ and $\delta\approx 10^{-7}$ were used for this experiment. Vector field colorings are obtained from a linear interpolation from red to blue independent of the vector fields or optimal flows.

Theorems & Definitions (31)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.1
• Proposition 2.1
• proof
• Corollary 2.1
• Definition 2.1
• Theorem 2.2