Optimal Transport for Measure and Image Interpolation and for Information Design

Abstract

In this work, we exploit the versatility and stability of optimal-transport–based methods to address challenges arising in computer vision, machine learning, and financial mathematics. <br/> Chapter 1 introduces a practical framework for spline interpolation over probability measures, utilizing the Riemannian geometry of Wasserstein space. The model defines distributional splines, continuous-time trajectories of distributions that balance smoothness and transport efficiency. We prove existence of minimizers and &Gamma;-convergence for selected discretizations. An efficient Nesterov-accelerated solver is presented, which scales to practical problem sizes and supports data modalities such as images and latent embeddings. This framework extends traditional machine learning techniques to data represented as probability distributions, including textures, single-cell genomics, and stochastic processes. We evaluate the model on three tasks: (i) generative texture video synthesis from a few exemplar frames, producing temporally coherent textures; (ii) latent-space interpolation with variational autoencoders (VAEs), yielding smooth, controllable interpolation and extrapolation paths; and (iii) Wasserstein regression with distribution-valued responses. Across these applications, the model demonstrates flexibility, stable optimization, and high qualitative performance. <br/> Chapter 2 generalizes this framework to unbalanced distributions, where the total mass can vary over time. We define a spline objective that combines three key components: (i) a diffeomorphic flow for geometric deformation, (ii) optimal transport for mass displacement, and (iii) a penalized source term to model mass creation and absorption. The resulting model jointly captures transport, deformation, and mass variation. Unlike the balanced case, this formulation is not Riemannian-consistent, but rather a pragmatic extension designed to handle scenarios with varying sample sizes and structural changes, common in computer vision and machine learning tasks. <br/> Chapter 3 applies optimal transport to a class of information design problems, focusing on learning optimal information policies. Using entropy-regularized optimal transport, solved via the Sinkhorn algorithm, we obtain scalable, numerically stable solutions. The entropy term introduces a temperature parameter that smooths the objective, improving conditioning and enabling fast matrix-vector-product iterations. As the temperature parameter ε approaches zero, the regularized solution converges to the original, unregularized problem. We demonstrate the approach through Bayesian persuasion in the context of the monopolist's problem, where strategic disclosure policies are learned to optimize decisions. The method shows that optimal information policies can improve expected utility relative to traditional approaches, even under varying dataset sizes and levels of model misspecification. Practical guidance is provided for deployment, including stochastic gradient descent and model selection routines such as early stopping and validation of &epsilon;.