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.
Highlights
Analysis of cross-sectional data is ubiquitous in machine learning and science
We chose Gaussian process (GP); neural networks would be well suited, We solve the aforementioned regression objectives using GPs [11] motivated by the connection between the drift of stochastic differential equations and GPs [12], We provide a conceptual comparison with the approach by [10] and detail why the density estimation formulation in [10] scales poorly with dimension, we re-implement the approach by [10] and compare approaches across a series of numerical experiments
We have presented iterative proportional maximum likelihood (IPML), a method to solve the Schrödinger bridge for arbitrary diffusion priors
Summary
Temporal data are typically sampled at discrete intervals due to technological or physical constraints. This means information between time points is lost. Schrödinger bridge problem [1,2] finds the most likely stochastic process that evolves a distribution π0 ( x) to another distribution π1 (y) consistently with a pre-specified Brownian motion. We consider a more general dynamical Schrödinger bridge problem for any pre-specified diffusion prior. This generalization allows us to exploit domain knowledge, e.g., oceanic and atmospheric flows might be interpolated from empirical measurements using previously established dynamics as priors
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