Schrödinger-Bass Bridge Barycenter Models for Measure-Valued Data

Ngoc-Hai Nguyen, Thuan Nguyen, Prakash Ishwar, Shuchin Aeron

Barycenter models, based on stochastic and deterministic optimal transport bridges between probability measures, have proven to be useful for supervised and unsupervised learning for (probability) measure-valued data. In this paper, we utilize the directional asymmetry and dual drift-diffusion control arising in the recently proposed Schrödinger-Bass Bridge (SBB) between two measures, and propose \textit{SBB-encoder}, \textit{SBB-decoder}, \textit{SBB-symmetric}, and \textit{SBB-Markovian} barycenter models. For each of these novel models, our first theoretical result establishes the existence of the SBB-barycenter under mild assumptions. Our second theoretical result characterizes the decoder SBB-barycenter of Gaussians where we show that for Gaussian input measures, the decoder formulation admits a Gaussian minimizer, reducing barycenter computation to a finite-dimensional optimization and yielding a compact algorithm. For general input measures, we derive a hybrid Wasserstein gradient particle descent scheme for all four formulations using the SBB dual representation. Finally, we utilize these algorithms for solving distributional-inverse/interpolation problems arising in the…