Schrödinger-Bass bridge models improve interpolation of measure-based data
Schrödinger-Bass Bridge Barycenter Models for Measure-Valued Data
Machine Learning
Summary
Sometimes we need to understand and work with data that is spread out in different shapes or distributions, like images or gene activity snapshots over time. This paper presents new mathematical models called Schrödinger-Bass Bridge barycenters to better combine and interpolate such complex data. The authors prove that these models work well theoretically and provide efficient algorithms to use them, especially when the data follows patterns like Gaussian distributions. They show that these methods can more accurately trace changes over time in areas like spatial images and gene sequencing than previous approaches.
What this means in practice
- •For image processing teams: Interpolate spatially-distributed image data across time points using improved measure barycenters for better temporal image reconstruction.
- •For bioinformatics developers: Infer intermediate states in single-cell RNA sequencing data to model biological progression with efficient Schrödinger-Bass bridge barycenter methods.
Authors
Ngoc-Hai Nguyen, Thuan Nguyen, Prakash Ishwar, Shuchin Aeron
Abstract
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 context of spatial images and single-cell RNA sequencing data, where we show that the SBB-decoder and Markovian barycenter allow us to interpolate between successive time points. Results show that our method outperforms existing barycenter models for measure interpolation and inference.