Department of Computer Science and Technology

Technical reports

Machine-learning approaches for the empirical Schrödinger bridge problem

Francisco Vargas

June 2021, 114 pages

This technical report is based on a dissertation submitted 24 June 2020 by the author for the degree of Master of Philosophy (Advanced Computer Science) to the University of Cambridge, Girton College.

This version of the report incorporates minor changes to the June 2021 original, which were released June 2022.

DOI: 10.48456/tr-958


The Schrödinger bridge problem is concerned with finding the most likely stochastic evolution between two probability distributions given a prior/reference stochastic evolution. This problem was posed by Schrödinger (1931, 1932) and solved to a large extent. Problems of this kind, whilst not popular in the machine learning community, have direct applications such as domain adaptation, hypothesis testing, semantic similarity, and others.

Thus, the focus of this thesis is to carry out a preliminary study on computational approaches for estimating the Schrödinger bridge between two distributions, when these distributions are available (or can be available) through samples, as most problems in machine learning are.

Due to the mathematical nature of the problem, this manuscript is also concerned with restating and re-deriving theorems and results that seem to be considered communal knowledge within the mathematical community or hidden in type-written textbooks behind paywalls. Part of the aim of this thesis is to make the mathematical machinery behind these approaches more accessible to a broader audience.

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BibTeX record

  author =	 {Vargas, Francisco},
  title = 	 {{Machine-learning approaches for the empirical
         	   Schr{\"o}dinger bridge problem}},
  year = 	 2021,
  month = 	 jun,
  url = 	 {},
  institution =  {University of Cambridge, Computer Laboratory},
  doi = 	 {10.48456/tr-958},
  number = 	 {UCAM-CL-TR-958}