Light and Optimal Schrödinger Bridge Matching (2402.03207v2)

Abstract: Schr\"odinger Bridges (SB) have recently gained the attention of the ML community as a promising extension of classic diffusion models which is also interconnected to the Entropic Optimal Transport (EOT). Recent solvers for SB exploit the pervasive bridge matching procedures. Such procedures aim to recover a stochastic process transporting the mass between distributions given only a transport plan between them. In particular, given the EOT plan, these procedures can be adapted to solve SB. This fact is heavily exploited by recent works giving rise to matching-based SB solvers. The cornerstone here is recovering the EOT plan: recent works either use heuristical approximations (e.g., the minibatch OT) or establish iterative matching procedures which by the design accumulate the error during the training. We address these limitations and propose a novel procedure to learn SB which we call the \textbf{optimal Schr\"odinger bridge matching}. It exploits the optimal parameterization of the diffusion process and provably recovers the SB process \textbf{(a)} with a single bridge matching step and \textbf{(b)} with arbitrary transport plan as the input. Furthermore, we show that the optimal bridge matching objective coincides with the recently discovered energy-based modeling (EBM) objectives to learn EOT/SB. Inspired by this observation, we develop a light solver (which we call LightSB-M) to implement optimal matching in practice using the Gaussian mixture parameterization of the adjusted Schr\"odinger potential. We experimentally showcase the performance of our solver in a range of practical tasks. The code for our solver can be found at https://github.com/SKholkin/LightSB-Matching.

• The paper introduces an optimal single-iteration approach to Schrödinger bridge matching, reducing error accumulation in dynamic optimal transport estimations.
• It leverages a Gaussian mixture parameterization to analytically express the Schrödinger bridge drift for rapid evaluation and training.
• Empirical results demonstrate robust performance across benchmarks, including low-dimensional mappings, high-dimensional biological data, and image-to-image translation.

Overview of "Light and Optimal Schrödinger Bridge Matching"

This paper introduces a novel approach for solving Schrödinger Bridges (SB), leveraging their connections to Entropic Optimal Transport (EOT). The authors propose a method termed "optimal Schrödinger bridge matching," which exhibits notable efficiency and efficacy in approximating SBs with a single bridge matching iteration. This addresses the inherent limitations of existing iterative approaches that accumulate errors throughout training.

Context and Motivation

Schrödinger Bridges have been a subject of interest because they extend classical diffusion models. The association of SBs with EOT provides a compelling dynamical viewpoint, where the optimal transport plan is not only a static mapping but part of a dynamic distribution transformation. Recent studies either approximate the EOT using heuristics, such as minibatch OT, or engage iterative techniques with accumulating error potentials. These methods, though useful, often introduce biases or are computationally inefficient.

Methodology

Optimal Schrödinger Bridge Matching

The authors' primary contribution is the "optimal Schrödinger bridge matching" procedure:

• Single Iteration Process: Rather than gradually improving an approximation, the proposed method achieves the SB with a single transition step through an optimal projection.
• Flexible Input Plans: This procedure’s efficacy holds regardless of the initial transport plan used, whether it be independently constructed, derived from minibatch OT, or otherwise.
• Gaussian Mixture Parameterization: An implementation cornerstone is the Gaussian mixture parameterization of the Schrödinger potential, facilitating analytical expression of the SB drift, permitting rapid evaluation and training.

Empirical Findings

A comprehensive suite of experiments demonstrates the methodology’s effectiveness across multiple standard tasks:

1. 2D Example: Efficiently maps distributions in low-dimensional spaces, adapting the level of noise as dictated by the transport regularization parameter, ε.
2. Benchmark Comparisons: On established benchmarks, the method shows competitive performance in estimating the true EOT plan and reconstructing target distributions, surpassing several leading bridge matching techniques.
3. Biological Data: In high-dimensional settings, such as single-cell expression data, the proposed method maintains robust performance and speed advantages.
4. Image-to-Image Translation: Successfully translates unpaired image datasets, showing applicability in broader domains requiring transformation without direct mapping availability.

Implications

Theoretical Impact: The provision of a singular iteration approach for SB solution represents a significant methodological advance, reducing computational burdens and mitigating error accumulation witnessed in iterative methods.

Practical Applications: By supporting arbitrary initial plans, the method allows broader usability across various domains, especially in situations with high-dimensional data or where computational resources may be constrained.

Future Directions

Given the advantages demonstrated, future research may focus on broadening the method’s applicability to other high-dimensional generative tasks. Moreover, exploration into richer parameterization schemes for the Schrödinger potential could maintain the method’s robustness while extending its applicability further into complex generative models, such as those encountered in modern AI systems.

In summary, this research not only refines existing methodologies for SB computation but also posits a scalable, efficient approach that advances the field’s understanding of dynamic optimal transport solutions.