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Diffusion Schrödinger Bridge Matching (2303.16852v3)

Published 29 Mar 2023 in stat.ML and cs.LG

Abstract: Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs) implement such a transport through a Stochastic Differential Equation (SDE) or an Ordinary Differential Equation (ODE). However, while it is desirable in many applications to approximate the deterministic dynamic Optimal Transport (OT) map which admits attractive properties, DDMs and FMMs are not guaranteed to provide transports close to the OT map. In contrast, Schr\"odinger bridges (SBs) compute stochastic dynamic mappings which recover entropy-regularized versions of OT. Unfortunately, existing numerical methods approximating SBs either scale poorly with dimension or accumulate errors across iterations. In this work, we introduce Iterative Markovian Fitting (IMF), a new methodology for solving SB problems, and Diffusion Schr\"odinger Bridge Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM significantly improves over previous SB numerics and recovers as special/limiting cases various recent transport methods. We demonstrate the performance of DSBM on a variety of problems.

Citations (56)

Summary

  • The paper introduces a novel Iterative Markovian Fitting approach that underpins Diffusion Schrödinger Bridge Matching for efficient optimal transport approximation.
  • It leverages stochastic dynamic mappings to solve entropy-regularized transport problems with improved scalability in complex, high-dimensional data spaces.
  • Experimental results demonstrate that DSBM outperforms traditional Denoising Diffusion Models and Flow Matching Models in benchmark transport tasks.

Overview of Diffusion Schrödinger Bridge Matching

This paper addresses the challenge of solving transport problems, specifically focusing on the construction of mappings to transform one probability distribution into another. Such transport problems are essential in various applications within machine learning, particularly in generative modeling. Conventional approaches include Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs), which realize transport through Stochastic Differential Equations (SDEs) or Ordinary Differential Equations (ODEs). However, these models may not always approximate the deterministic dynamic Optimal Transport (OT) efficiently. Instead, Schrödinger Bridges (SBs) offer a promising alternative by enabling stochastic dynamic mappings that solve entropy-regularized OT problems, albeit with computational challenges in high-dimensional settings.

Iterative Markovian Fitting and Diffusion Schrödinger Bridge Matching

The paper introduces a novel approach dubbed Iterative Markovian Fitting (IMF) aimed at computing SBs more effectively. This methodology iteratively approaches the SB solution by alternating projections onto the spaces of Markovian and reciprocal classes, refining the approximation of the SB at each step. Diffusion Schrödinger Bridge Matching (DSBM) is proposed as a practical algorithmic framework that applies IMF to effectively solve transport tasks. By employing DSBM, the authors claim significant improvements in solving SB problems, demonstrating its utility across various numerical and experimental settings.

Strong Numerical Results and Experimental Findings

The research provides robust numerical results, showcasing the effectiveness of DSBM in solving complex transport tasks. Notably, DSBM efficiently handles the intricacies arising from high-dimensional data, which have traditionally imposed scalability issues on other SB approximation techniques. In comparative benchmarks with existing models, DSBM achieves commendable performance metrics, thereby substantiating its computational advantages over existing SB frameworks.

Theoretical and Practical Implications

The introduction of DSBM and IMF holds both theoretical and practical implications. Theoretically, these models introduce a framework for precisely approximating solutions to entropy-regularized OT, which could advance the understanding of dynamic process modeling. Practically, the approach promises improvements in generative model design, potentially enhancing tasks such as image generation, style transfer, and more. This could lead to further exploration of dynamic mass transport problems in machine learning and broader AI advancements.

Future Directions

Future research could explore refining the proposed methodologies to further reduce computational complexity, especially in extremely high-dimensional tasks. Additionally, integrating these approaches with emerging AI technologies could unveil broader applications, potentially transforming how complex data mapping tasks are approached across different domains. As the AI field evolves, the insights gathered from this research could guide the development of more sophisticated models that better emulate natural processes and complex systems.

This paper provides a comprehensive contribution to the field of computational transport and has potential implications for advancing both theoretical research and practical applications in artificial intelligence.