Solving Schrödinger Bridges via Maximum Likelihood

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.

[1]  I. Gyöngy,et al.  Existence of strong solutions for Itô's stochastic equations via approximations , 1996 .

[2]  Diederik P. Kingma,et al.  Variational Diffusion Models , 2021, ArXiv.

[3]  Yuling Jiao,et al.  Schr{\"o}dinger-F{\"o}llmer Sampler: Sampling without Ergodicity , 2021, 2106.10880.

[4]  Gefei Wang,et al.  Deep Generative Learning via Schrödinger Bridge , 2021, ICML.

[5]  Valentin De Bortoli,et al.  Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling , 2021, NeurIPS.

[6]  Michele Pavon,et al.  The Data‐Driven Schrödinger Bridge , 2021, Communications on Pure and Applied Mathematics.

[7]  Abhishek Kumar,et al.  Score-Based Generative Modeling through Stochastic Differential Equations , 2020, ICLR.

[8]  Lénaïc Chizat,et al.  Faster Wasserstein Distance Estimation with the Sinkhorn Divergence , 2020, NeurIPS.

[9]  D. V. Dijk,et al.  TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics , 2020, ICML.

[10]  George Papamakarios,et al.  Neural Density Estimation and Likelihood-free Inference , 2019, ArXiv.

[11]  P. Rigollet,et al.  Optimal-Transport Analysis of Single-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming , 2019, Cell.

[12]  P. Rigollet,et al.  Optimal-Transport Analysis of Single-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming , 2019, Cell.

[13]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Alain Trouvé,et al.  Interpolating between Optimal Transport and MMD using Sinkhorn Divergences , 2018, AISTATS.

[15]  Andreas Ruttor,et al.  Approximate Bayes learning of stochastic differential equations. , 2017, Physical review. E.

[16]  Iain Murray,et al.  Masked Autoregressive Flow for Density Estimation , 2017, NIPS.

[17]  Luca Martino,et al.  Effective sample size for importance sampling based on discrepancy measures , 2016, Signal Process..

[18]  Michael A. Osborne,et al.  Probabilistic numerics and uncertainty in computations , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  Max Welling,et al.  Auto-Encoding Variational Bayes , 2013, ICLR.

[20]  Christian L'eonard.  Some properties of path measures , 2013, 1308.0217.

[21]  Marco Cuturi,et al.  Sinkhorn Distances: Lightspeed Computation of Optimal Transport , 2013, NIPS.

[22]  Christian L'eonard.  A survey of the Schr\"odinger problem and some of its connections with optimal transport , 2013, 1308.0215.

[23]  Andreas Ruttor,et al.  Approximate Gaussian process inference for the drift function in stochastic differential equations , 2013, NIPS.

[24]  Suvrit Sra,et al.  Scalable nonconvex inexact proximal splitting , 2012, NIPS.

[25]  G. Roberts,et al.  Nonparametric estimation of diffusions: a differential equations approach , 2012 .

[26]  Neil D. Lawrence,et al.  Kernels for Vector-Valued Functions , 2012 .

[27]  A. M. Stuart,et al.  Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs , 2012, 1202.0976.

[28]  Neil D. Lawrence,et al.  Kernels for Vector-Valued Functions: a Review , 2011, Found. Trends Mach. Learn..

[29]  Nestor Guillen,et al.  Five lectures on optimal transportation: Geometry, regularity and applications , 2010, 1011.2911.

[30]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[31]  Bernard C. Levy,et al.  Principles of Signal Detection and Parameter Estimation , 2008 .

[32]  S. Sarkka,et al.  Application of Girsanov Theorem to Particle Filtering of Discretely Observed Continuous - Time Non-Linear Systems , 2007, 0705.1598.

[33]  Neil D. Lawrence,et al.  Probabilistic inference of transcription factor concentrations and gene-specific regulatory activities , 2006, Bioinform..

[34]  Charles A. Micchelli,et al.  Learning Multiple Tasks with Kernel Methods , 2005, J. Mach. Learn. Res..

[35]  Gramer Erhard.  PROBABILITY MEASURES WITH GIVEN MARGINALS AND CONDITIONALS: I-PROJECTIONS AND CONDITIONAL ITERATIVE PROPORTIONAL FITTING , 2000 .

[36]  L. Rüschendorf.  Convergence of the iterative proportional fitting procedure , 1995 .

[37]  M. Pavon,et al.  On Free Energy, Stochastic Control, and Schrödinger Processes , 1991 .

[38]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[39]  E. Pardoux,et al.  Stochastic Differential Systems: Filtering and Control , 1985 .

[40]  Brian D. O. Anderson,et al.  Reverse time diffusions , 1985 .

[41]  Hans Föllmer,et al.  An entropy approach to the time reversal of diffusion processes , 1985 .

[42]  U. Haussmann,et al.  Time reversal of diffusion processes , 1985 .

[43]  B. Anderson.  Reverse-time diffusion equation models , 1982 .

[44]  Hiroshi Kunitha.  On backward stochastic differential equations , 1982 .

[45]  T. Kailath.  The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures , 1971 .

[46]  S. Kullback.  Probability Densities with Given Marginals , 1968 .

[47]  Richard Sinkhorn,et al.  Concerning nonnegative matrices and doubly stochastic matrices , 1967 .

[48]  Edward Nelson.  Dynamical Theories of Brownian Motion , 1967 .

[49]  E. Schrödinger.  Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique , 1932 .

[50]  H. P..  Annales de l'Institut Henri Poincaré , 1931, Nature.