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BM$^2$: Coupled Schrödinger Bridge Matching

Stefano Peluchetti

TL;DR

BM$^2$ introduces a non-iterative neural method to learn Schrödinger bridges between two target distributions under a tractable reference diffusion. By jointly parameterizing the forward and backward drifts with a single neural network and training with a pair of drift-matching losses, BM$^2$ achieves an exact bridge in the idealized limit and practical accuracy limited only by function approximation and discretization. The authors establish convergence motifs, including that the SB solution $S$ is a fixed point for the coupled system and discuss complete, partial, and infinitesimal minimization regimes, showing links to IPF-like dynamics. Empirically, BM$^2$ outperforms iterative baselines like I-BM and DIPF across dimensions and entropic regularization levels on standard benchmarks, while remaining memory-efficient and straightforward to implement for diffusion-based generative tasks.

Abstract

A Schrödinger bridge establishes a dynamic transport map between two target distributions via a reference process, simultaneously solving an associated entropic optimal transport problem. We consider the setting where samples from the target distributions are available, and the reference diffusion process admits tractable dynamics. We thus introduce Coupled Bridge Matching (BM$^2$), a simple non-iterative approach for learning Schrödinger bridges with neural networks. A preliminary theoretical analysis of the convergence properties of BM$^2$ is carried out, supported by numerical experiments that demonstrate the effectiveness of our proposal.

BM$^2$: Coupled Schrödinger Bridge Matching

TL;DR

BM introduces a non-iterative neural method to learn Schrödinger bridges between two target distributions under a tractable reference diffusion. By jointly parameterizing the forward and backward drifts with a single neural network and training with a pair of drift-matching losses, BM achieves an exact bridge in the idealized limit and practical accuracy limited only by function approximation and discretization. The authors establish convergence motifs, including that the SB solution is a fixed point for the coupled system and discuss complete, partial, and infinitesimal minimization regimes, showing links to IPF-like dynamics. Empirically, BM outperforms iterative baselines like I-BM and DIPF across dimensions and entropic regularization levels on standard benchmarks, while remaining memory-efficient and straightforward to implement for diffusion-based generative tasks.

Abstract

A Schrödinger bridge establishes a dynamic transport map between two target distributions via a reference process, simultaneously solving an associated entropic optimal transport problem. We consider the setting where samples from the target distributions are available, and the reference diffusion process admits tractable dynamics. We thus introduce Coupled Bridge Matching (BM), a simple non-iterative approach for learning Schrödinger bridges with neural networks. A preliminary theoretical analysis of the convergence properties of BM is carried out, supported by numerical experiments that demonstrate the effectiveness of our proposal.
Paper Structure (21 sections, 10 theorems, 34 equations, 2 figures, 3 tables, 2 algorithms)

This paper contains 21 sections, 10 theorems, 34 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Schrödinger Bridges and Entropic Optimal Transport
  4. Reference Dynamics
  5. Bridge Matching (BM)
  6. Iterated Bridge Matching (I-BM) and Diffusion Iterative Proportional Fitting (DIPF)
  7. Coupled BM (BM$^2$)
  8. Implementation Aspects
  9. Convergence Properties
  10. Complete Minimization
  11. Partial Minimization
  12. Infinitesimal Minimization
  13. Numerical Experiments
  14. Related Works
  15. Conclusions
  16. ...and 6 more sections

Key Result

Lemma 1

Under suitable conditions leonard2014properties, the updates $(H',K') \overset{eq:bm2_system}{→} (H^{K'_{0,1}},K^{H'_{0,1}})$, parametrized by diffusion process distributions, admit $H' = K' = S$ as fixed point. If $H' = K'$, this fixed point is unique.

Figures (2)

  • Figure 1: Evolution of the metric \ref{['eq:metric_cbw']} for $d=64$, $ε=1$ over SGD steps for BM$^2$, I-BM and DIPF.
  • Figure 2: Algorithmic-time $l$ evolution of $𝔼_{F^{(l)}}[X_1]$, $𝕍_{F^{(l)}}[X_1]$, $ℂ_{F^{(l)}}[X_0,X_1]$, compared with $𝔼_S[X_1]$, $𝕍_S[X_1]$, $ℂ_S[X_0,X_1]$ as dashed gray lines.

Theorems & Definitions (14)

  • Lemma 1: Fixed points of \ref{['eq:bm2_system']}
  • Theorem 1: Complete BM$^2$ Iterations
  • Lemma 2: Loss Interpretation
  • Theorem 2: Partial BM$^2$ Iterations
  • Lemma 3: $ℛ$-stability of $F^{(λ)}, B^{(λ)}$
  • Lemma 4: $𝒮$-stability of $F^{(λ)}, B^{(λ)}$
  • Theorem 2: Complete BM$^2$ Iterations
  • proof
  • Lemma 4: Loss Interpretation
  • proof
  • ...and 4 more