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Motivated by approximation Bayesian computation using mean-field variational approximation and the computation of equilibrium in multi-species systems with cross-interaction, this paper investigates the composite geodesically convex optimization problem over multiple distributions. The objective functional under consideration is composed of a convex potential energy on a product of Wasserstein spaces and a sum of convex self-interaction and internal energies associated with each distribution. To efficiently solve this problem, we introduce the Wasserstein Proximal Coordinate Gradient (WPCG) algorithms with parallel, sequential, and random update schemes. Under a quadratic growth (QG) condition that is weaker than the usual strong convexity requirement on the objective functional, we show that WPCG converges exponentially fast to the unique global optimum. In the absence of the QG condition, WPCG is still demonstrated to converge to the global optimal solution, albeit at a slower polynomial rate. Numerical results for both motivating examples are consistent with our theoretical findings. 

Optimal transport (OT) offers a versatile framework to compare complex data distributions in a geometrically meaningful way. Traditional methods for computing the Wasserstein distance and geodesic between probability measures require mesh-specific domain discretization and suffer from the curse-of-dimensionality. We present GeONet, a mesh-invariant deep neural operator network that learns the non-linear mapping from the input pair of initial and terminal distributions to the Wasserstein geodesic connecting the two endpoint distributions. In the offline training stage, GeONet learns the saddle point optimality conditions for the dynamic formulation of the OT problem in the primal and dual spaces that are characterized by a coupled PDE system. The subsequent inference stage is instantaneous and can be deployed for real-time predictions in the online learning setting. We demonstrate that GeONet achieves comparable testing accuracy to the standard OT solvers on simulation examples and the MNIST dataset with considerably reduced inference-stage computational cost by orders of magnitude.