A Score-Based Approach for Training Schrödinger Bridges for Data Modelling

Ludwig Winkler; Cesar Ojeda; Manfred Opper

doi:10.3390/e25020316

A Score-Based Approach for Training Schrödinger Bridges for Data Modelling

Entropy (Basel). 2023 Feb 8;25(2):316. doi: 10.3390/e25020316.

Authors

Ludwig Winkler¹, Cesar Ojeda^{2

3}, Manfred Opper^{2

3

4}

Affiliations

¹ Machine Learning Group, Technische Universität Berlin, 10623 Berlin, Germany.
² Artificial Intelligence Group, Technische Universität Berlin, 10623 Berlin, Germany.
³ Institute of Mathematics, University of Potsdam, 14469 Potsdam, Germany.
⁴ Centre for Systems Modelling and Quantitative Biomedicine, University of Birmingham, Birmingham B15 2TT, UK.

Abstract

A Schrödinger bridge is a stochastic process connecting two given probability distributions over time. It has been recently applied as an approach for generative data modelling. The computational training of such bridges requires the repeated estimation of the drift function for a time-reversed stochastic process using samples generated by the corresponding forward process. We introduce a modified score- function-based method for computing such reverse drifts, which can be efficiently implemented by a feed-forward neural network. We applied our approach to artificial datasets with increasing complexity. Finally, we evaluated its performance on genetic data, where Schrödinger bridges can be used to model the time evolution of single-cell RNA measurements.

Keywords: Schrödinger bridge problem; reverse-time stochastic processes; score estimation.

Grants and funding

M.O. and C.O. were partially funded by Deutsche Forschungsgemeinschft (DFG) Project-ID 318763901-SFB1294. The research of L.W. was funded by the BIFOLD-Berlin Institute for the Foundations of Learning and Data (Reference 01IS18025A and Reference 01IS18037A).