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Plug-in estimation of Schrödinger bridges

Aram-Alexandre Pooladian, Jonathan Niles-Weed

TL;DR

This work introduces the Sinkhorn bridge, a plug-in estimator for the Schrödinger bridge between two distributions that leverages static entropic OT potentials computed from samples. By expressing the SB drift via heat semigroups and using the optimal potentials from a single Sinkhorn run, the method avoids forward/backward diffusion training and neural nets, while providing statistical guarantees that depend on the target's intrinsic dimension. Theoretical results establish one-sample convergence rates and discretization errors, with specialized results for the Föllmer bridge and practical sampling implications. Numerical experiments in Gaussian and multimodal regimes corroborate the favorable sample efficiency and accuracy of the Sinkhorn bridge relative to existing neural-sde-based approaches. Overall, the paper links entropic OT, SB, and Sinkhorn methods to deliver a scalable, provably accurate estimator for dynamic transport between distributions.

Abstract

We propose a procedure for estimating the Schrödinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schrödinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport.

Plug-in estimation of Schrödinger bridges

TL;DR

This work introduces the Sinkhorn bridge, a plug-in estimator for the Schrödinger bridge between two distributions that leverages static entropic OT potentials computed from samples. By expressing the SB drift via heat semigroups and using the optimal potentials from a single Sinkhorn run, the method avoids forward/backward diffusion training and neural nets, while providing statistical guarantees that depend on the target's intrinsic dimension. Theoretical results establish one-sample convergence rates and discretization errors, with specialized results for the Föllmer bridge and practical sampling implications. Numerical experiments in Gaussian and multimodal regimes corroborate the favorable sample efficiency and accuracy of the Sinkhorn bridge relative to existing neural-sde-based approaches. Overall, the paper links entropic OT, SB, and Sinkhorn methods to deliver a scalable, provably accurate estimator for dynamic transport between distributions.

Abstract

We propose a procedure for estimating the Schrödinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schrödinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport.
Paper Structure (32 sections, 14 theorems, 112 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 32 sections, 14 theorems, 112 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. On Schrödinger bridges.
  4. Background
  5. Entropic optimal transport
  6. Static formulation
  7. A dynamic formulation via the continuity equation
  8. A stochastic formulation via the Fokker--Planck equation
  9. The Schrödinger Bridge problem
  10. Proposed estimator: The Sinkhorn bridge
  11. From Schrödinger to Sinkhorn and back
  12. Defining the estimator
  13. Main results and proof sketch
  14. Statistical analysis
  15. Completing the results
  16. ...and 17 more sections

Key Result

Proposition 3.1

Let $\pi$ be a probability measure of the form for any measurable $f$ and $g$ and any probability measures $\mu_0,\mu_1 \in \mathcal{P}_2(\mathbb{R}^d)$. Let $\mathsf{M}$ the path measure given by a mixture of Brownian bridges with respect to eq:pi_generic as in eq:sb_first, with $t$-marginals $\mathsf{m}_t$ for $t \in [0,1]$. The following hold

Figures (3)

  • Figure 1: Schrödinger bridges on the basis of samples from toy datasets.
  • Figure 2: MSE for estimating the Gaussian drift as $(n,\tau)$ vary, averaged over 10 trials.
  • Figure 3: Plotting generated and resampled target data in $d=64$.

Theorems & Definitions (27)

  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1: One-sample estimation; no discretization
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Theorem 4.5
  • Corollary 4.6
  • Remark 4.7
  • ...and 17 more