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Solving Schrödinger Bridges via Maximum Likelihood
Francisco Vargas1, Pierre Thodoroff1, Austen Lamacraft2
1The Computer Laboratory, Department of Computer Science and Technology, University of Cambridge, William Gates Building, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK.
Entropy (Basel, Switzerland)
|September 28, 2021
Summary
The Schrödinger bridge problem (SBP) can now be solved using machine learning techniques. Our new method connects SBP to autoregressive maximum likelihood estimation, enabling scalable numerical solutions for probability distribution evolution.
Area of Science:
- Computational statistics
- Machine learning
- Probability theory
Background:
- The Schrödinger bridge problem (SBP) models the most probable stochastic process between two probability distributions.
- While theoretically understood, efficient numerical methods for SBP estimation are lacking.
- SBP has applications in natural sciences and machine learning, including dataset alignment and hypothesis testing.
Purpose of the Study:
- To develop a scalable numerical method for estimating Schrödinger bridges.
- To establish a connection between SBP and machine learning objectives.
- To demonstrate the practical utility of the proposed approach.
Main Methods:
- Proving the equivalence between solving the SBP and an autoregressive maximum likelihood estimation (MLE) objective.
- Developing a numerical procedure using Gaussian processes for SBP estimation.
- Validating the method through numerical simulations and experiments.
Main Results:
- Demonstrated equivalence between SBP and autoregressive MLE, bypassing density estimation challenges.
- Proposed a novel Gaussian process-based numerical procedure for SBP estimation.
- Successfully applied the method in numerical simulations and experiments, showcasing its practical viability.
Conclusions:
- The proposed formulation enables direct application of advanced machine learning techniques to SBP.
- The Gaussian process-based method offers a scalable and practical approach to estimating Schrödinger bridges.
- This work bridges theoretical SBP with practical machine learning implementations, opening new avenues for research and application.
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