Solving Schrödinger Bridges via Maximum Likelihood

Francisco Vargas; Pierre Thodoroff; Austen Lamacraft; Neil D. Lawrence

doi:10.3390/e23091134

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Solving Schrödinger Bridges via Maximum Likelihood

Francisco Vargas, Pierre Thodoroff, Austen Lamacraft, Neil D. Lawrence

Entropy, 23(9):1134, 2021.

Abstract

The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.

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BibTeX


@Article{Vargas-solving21,
  title = 	 {Solving Schrödinger Bridges via Maximum Likelihood},
  author = 	 {Vargas, Francisco and Thodoroff, Pierre and Lamacraft, Austen and Lawrence, Neil D.},
  journal =      {Entropy},
  year = 	 {2021},
  volume = 	 {23},
  number =       {9},
  doi = 	 {10.3390/e23091134},
  pdf = 	 {https://www.mdpi.com/1099-4300/23/9/1134/pdf},
  url = 	 {/publications/solving-schroedinger-bridges-via-maximum-likelihood.html},
  abstract = 	 {The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.
}
}

Endnote

%0 Journal Article
%T Solving Schrödinger Bridges via Maximum Likelihood
%A Francisco Vargas
%A Pierre Thodoroff
%A Austen Lamacraft
%A Neil D. Lawrence
%J Entropy
%D 2021	
%F Vargas-solving21
%R 10.3390/e23091134
%U /publications/solving-schroedinger-bridges-via-maximum-likelihood.html
%V 23
%N 9
%X The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.

RIS


TY  - JOUR
TI  - Solving Schrödinger Bridges via Maximum Likelihood
AU  - Francisco Vargas
AU  - Pierre Thodoroff
AU  - Austen Lamacraft
AU  - Neil D. Lawrence
DA  - 2021/08/31	
ID  - Vargas-solving21
VL  - 23
IS  - 9
SP  - 1134
DO  - 10.3390/e23091134
L1  - https://www.mdpi.com/1099-4300/23/9/1134/pdf
UR  - /publications/solving-schroedinger-bridges-via-maximum-likelihood.html
AB  - The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.

ER  -

APA


Vargas, F., Thodoroff, P., Lamacraft, A. & Lawrence, N.D.. (2021). Solving Schrödinger Bridges via Maximum Likelihood. Entropy 23(9):1134 doi:10.3390/e23091134 Available from /publications/solving-schroedinger-bridges-via-maximum-likelihood.html.