Schrödinger Bridge Flow for Unpaired Data Translation (2409.09347v1)

Abstract: Mass transport problems arise in many areas of machine learning whereby one wants to compute a map transporting one distribution to another. Generative modeling techniques like Generative Adversarial Networks (GANs) and Denoising Diffusion Models (DDMs) have been successfully adapted to solve such transport problems, resulting in CycleGAN and Bridge Matching respectively. However, these methods do not approximate Optimal Transport (OT) maps, which are known to have desirable properties. Existing techniques approximating OT maps for high-dimensional data-rich problems, such as DDM-based Rectified Flow and Schr\"odinger Bridge procedures, require fully training a DDM-type model at each iteration, or use mini-batch techniques which can introduce significant errors. We propose a novel algorithm to compute the Schr\"odinger Bridge, a dynamic entropy-regularised version of OT, that eliminates the need to train multiple DDM-like models. This algorithm corresponds to a discretisation of a flow of path measures, which we call the Schr\"odinger Bridge Flow, whose only stationary point is the Schr\"odinger Bridge. We demonstrate the performance of our algorithm on a variety of unpaired data translation tasks.

• The paper introduces a novel algorithm that discretizes the Schrödinger Bridge Flow to compute optimal transport maps without retraining multiple models.
• The work details the efficient α-IMF and α-DSBM procedures that converge to the Schrödinger Bridge, reducing cumulative errors in high-dimensional settings.
• Empirical results on image translation tasks demonstrate improved visual quality and accuracy compared to existing state-of-the-art methods.

Schrödinger Bridge Flow for Unpaired Data Translation

The paper "Schrödinger Bridge Flow for Unpaired Data Translation" addresses the problem of mass transport in machine learning, particularly focusing on the challenge of computing maps that transport one probability distribution to another. While methods such as Generative Adversarial Networks (GANs) and Denoising Diffusion Models (DDMs) have been adapted to these applications, existing approaches do not approximate Optimal Transport (OT) maps, which possess desirable theoretical properties.

The authors introduce a new algorithm to compute the Schrödinger Bridge (SB) for high-dimensional data. This procedure does not require repeatedly training DDM-type models or the use of mini-batch techniques, which can introduce significant errors due to high dimensionality. The core idea is to discretize a flow of path measures known as the Schrödinger Bridge Flow, where the stationary point ideally corresponds to the Schrödinger Bridge.

Key Contributions

1. Novel Algorithm for Schrödinger Bridge: The paper proposes an efficient computational algorithm to find the SB without needing to train multiple DDM-like models. This method discretizes a trajectory of path measures for which the stationary points are Schrödinger Bridges. By leveraging properties of Markov processes and the reciprocal class of Brownian motions, the authors introduce a novel approach termed the Schrödinger Bridge Flow.
2. The Schrödinger Bridge Flow: Theoretical underpinnings for the flow connect to the likelihood framework under standard OT and entropic OT (EOT). The technique shows that a certain class of path measures can approximate SB in a highly efficient manner compared to existing methods.
3. $\alpha$-IMF and $\alpha$-DSBM Procedures: An important innovation is the introduction of the $\alpha$-Iterative Markovian Fitting (IMF) procedure. This discretized flow algorithm generalizes the classic IMF, converging towards the SB for any discretization parameter $\alpha$ in the interval $(0, 1]$. Continued refinements lead to an online adaptation called $\alpha$-DSBM (Diffusion Schrödinger Bridge Matching), which features incremental learning steps.
4. Theoretical and Empirical Validation: The paper provides rigorous proofs demonstrating how $\alpha$-IMF and its variants converge to the SB. The effectiveness of these methods is empirically validated through several experiments on unpaired data translation tasks.

Numerical Results

In practical evaluations, the novel algorithm's performance was demonstrated on image-to-image translation tasks. Results showed that the proposed Schrödinger Bridge flow algorithm achieved high-accuracy transformations, improving upon both visual quality and alignment of results when compared to existing state-of-the-art methods.

Comparison and Implications

This approach eliminates the costly need to retrain generative models for each iteration, addressing a significant bottleneck in current state-of-the-art techniques. By maintaining computational efficiency and reducing cumulative errors via the introduction of $\alpha$-IMF, this work establishes a new benchmark for the OT problem under the Schrödinger Bridge framework.

From a theoretical standpoint, this paper extends existing knowledge in high-dimensional transport problems, framing these solutions within the regularized dynamic OT framework. The practical benefits are echoed in potential advancements and applications in computer vision and generative modeling, particularly in areas with large unpaired datasets.

Future Developments

Looking ahead, the implications of the Schrödinger Bridge Flow algorithm can extend beyond generative models alone. Future work could investigate its applications in reinforcement learning, where efficient exploration strategies are crucial, and in financial modeling, where dynamic OT interpretations provide robust risk measures. Given the scalable nature of this algorithm, it opens up pathways to further investigate self-consistency conditions and potential hybrid models that balance efficiency with theoretical guarantees.

In summary, this paper makes substantial contributions to the field of generative modeling and optimal transport by introducing an efficient and practical method to compute Schrödinger Bridges, blending insights from theoretical transport problems with practical, empirical methodologies. The incremental improvements brought by the $\alpha$-IMF and $\alpha$-DSBM procedures support a broad range of future applications in various domains of machine learning and data science.