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Feedback Schrödinger Bridge Matching

Panagiotis Theodoropoulos, Nikolaos Komianos, Vincent Pacelli, Guan-Horng Liu, Evangelos A. Theodorou

TL;DR

Feedback Schrödinger Bridge Matching (FSBM) presents a semi-supervised framework for distribution matching that injects state feedback from a small set of pre-aligned pairs into Schrödinger Bridge/Entropic OT formulations. By converting a static semi-supervised objective into a dynamic, two-stage optimization—intermediate path refinement and drift coupling—FSBM leverages partial supervision to guide unpaired samples, achieving faster training and stronger generalization across tasks. Empirical results in crowd navigation, opinion dynamics, and image translation show that FSBM can outperform fully unsupervised or fully supervised baselines with modest computational overhead. This approach broadens the practicality of diffusion-bridge matching by effectively utilizing partially aligned data to shape transport maps in high-dimensional settings.

Abstract

Recent advancements in diffusion bridges for distribution transport problems have heavily relied on matching frameworks, yet existing methods often face a trade-off between scalability and access to optimal pairings during training. Fully unsupervised methods make minimal assumptions but incur high computational costs, limiting their practicality. On the other hand, imposing full supervision of the matching process with optimal pairings improves scalability, however, it can be infeasible in many applications. To strike a balance between scalability and minimal supervision, we introduce Feedback Schrödinger Bridge Matching (FSBM), a novel semi-supervised matching framework that incorporates a small portion (less than 8% of the entire dataset) of pre-aligned pairs as state feedback to guide the transport map of non coupled samples, thereby significantly improving efficiency. This is achieved by formulating a static Entropic Optimal Transport (EOT) problem with an additional term capturing the semi-supervised guidance. The generalized EOT objective is then recast into a dynamic formulation to leverage the scalability of matching frameworks. Extensive experiments demonstrate that FSBM accelerates training and enhances generalization by leveraging coupled pairs guidance, opening new avenues for training matching frameworks with partially aligned datasets.

Feedback Schrödinger Bridge Matching

TL;DR

Feedback Schrödinger Bridge Matching (FSBM) presents a semi-supervised framework for distribution matching that injects state feedback from a small set of pre-aligned pairs into Schrödinger Bridge/Entropic OT formulations. By converting a static semi-supervised objective into a dynamic, two-stage optimization—intermediate path refinement and drift coupling—FSBM leverages partial supervision to guide unpaired samples, achieving faster training and stronger generalization across tasks. Empirical results in crowd navigation, opinion dynamics, and image translation show that FSBM can outperform fully unsupervised or fully supervised baselines with modest computational overhead. This approach broadens the practicality of diffusion-bridge matching by effectively utilizing partially aligned data to shape transport maps in high-dimensional settings.

Abstract

Recent advancements in diffusion bridges for distribution transport problems have heavily relied on matching frameworks, yet existing methods often face a trade-off between scalability and access to optimal pairings during training. Fully unsupervised methods make minimal assumptions but incur high computational costs, limiting their practicality. On the other hand, imposing full supervision of the matching process with optimal pairings improves scalability, however, it can be infeasible in many applications. To strike a balance between scalability and minimal supervision, we introduce Feedback Schrödinger Bridge Matching (FSBM), a novel semi-supervised matching framework that incorporates a small portion (less than 8% of the entire dataset) of pre-aligned pairs as state feedback to guide the transport map of non coupled samples, thereby significantly improving efficiency. This is achieved by formulating a static Entropic Optimal Transport (EOT) problem with an additional term capturing the semi-supervised guidance. The generalized EOT objective is then recast into a dynamic formulation to leverage the scalability of matching frameworks. Extensive experiments demonstrate that FSBM accelerates training and enhances generalization by leveraging coupled pairs guidance, opening new avenues for training matching frameworks with partially aligned datasets.

Paper Structure

This paper contains 38 sections, 5 theorems, 74 equations, 16 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schrödinger Bridges
  4. Schrödinger Bridge Matching
  5. Methodology
  6. Semi-Supervised Guidance
  7. Feedback Schrödinger Bridge Matching
  8. Dynamic objective
  9. Intermediate Path Optimization
  10. Feedback
  11. Intemediate Path
  12. Coupling Optimization
  13. Training Scheme
  14. Experiments
  15. Crowd Navigation
  16. ...and 23 more sections

Key Result

Theorem 3.2

Assume $X_0, X_1 \in{\mathbb{R}}^d$, with $X_0\sim\pi_0$, and $X_1\sim\pi_1$, and consider a stochastic random variable $X_t$ connecting $X_0$ and $X_1$, whose law is the continuous marginal probability path $p_t$ joining $\pi_0, \text{ and } \pi_1$. Additionally consider the KP pairs $\{x_0^n, x_1^

Figures (16)

  • Figure 1: FSBM connecting existing Bridge Matching frameworks at the extremes, where the dataset is comprised of either fully aligned or fully non-aligned pairs
  • Figure 2: Source and Target Distributions with their KP set pairs
  • Figure 3: Gradient Field of $\nabla G$ in the S-tunnel
  • Figure 4: Crowd Navigation comparison between GSBM and FSBM in the S-tunnel, and V-neck under perturbed mean and uniform initial distribution, and in the V-neck under perturbed mean and perturbed STD. The color of particles means: i) Yellow: initial conditions, ii) Green and Cyan: intermediate trajectory, iii) Red: target distribution, iv) Navy Blue: generated distribution
  • Figure 5: ${\mathcal{W}}_2$ distance between generated and ground truth distributions for GSBM and FSBM in (Lower is better)
  • ...and 11 more figures

Theorems & Definitions (11)

  • Theorem 3.2
  • Proposition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 3.6
  • proof
  • proof
  • proof
  • proof
  • Proposition A.1
  • ...and 1 more