Geodesics in fibered latent spaces: A geometric approach to learning correspondences between conditions

TL;DR

This paper presents a geometric framework using fibered latent spaces and a novel neural network architecture to learn correspondences between different conditions, with applications in batch correction and dataset integration.

Contribution

The work introduces a fiber bundle-based latent space formalism and a new network architecture for learning condition correspondences with a geometric approach.

Findings

Effective in batch correction tasks

Achieves diffeomorphism-based correspondences

Benchmarked on MNIST and Olivetti datasets

Abstract

This work introduces a geometric framework and a novel network architecture for creating correspondences between samples of different conditions. Under this formalism, the latent space is a fiber bundle stratified into a base space encoding conditions, and a fiber space encoding the variations within conditions. Furthermore, this latent space is endowed with a natural pull-back metric. The correspondences between conditions are obtained by minimizing an energy functional, resulting in diffeomorphism flows between fibers. We illustrate this approach using MNIST and Olivetti and benchmark its performances on the task of batch correction, which is the problem of integrating multiple biological datasets together.