Variational Schrödinger Diffusion Models (2405.04795v4)

Abstract: Schr\"odinger bridge (SB) has emerged as the go-to method for optimizing transportation plans in diffusion models. However, SB requires estimating the intractable forward score functions, inevitably resulting in the costly implicit training loss based on simulated trajectories. To improve the scalability while preserving efficient transportation plans, we leverage variational inference to linearize the forward score functions (variational scores) of SB and restore simulation-free properties in training backward scores. We propose the variational Schr\"odinger diffusion model (VSDM), where the forward process is a multivariate diffusion and the variational scores are adaptively optimized for efficient transport. Theoretically, we use stochastic approximation to prove the convergence of the variational scores and show the convergence of the adaptively generated samples based on the optimal variational scores. Empirically, we test the algorithm in simulated examples and observe that VSDM is efficient in generations of anisotropic shapes and yields straighter sample trajectories compared to the single-variate diffusion. We also verify the scalability of the algorithm in real-world data and achieve competitive unconditional generation performance in CIFAR10 and conditional generation in time series modeling. Notably, VSDM no longer depends on warm-up initializations and has become tuning-friendly in training large-scale experiments.

• The paper presents a novel variational approach that linearizes complex functions, eliminating the need for simulated trajectories in diffusion model training.
• The paper validates its method with theoretical convergence guarantees using stochastic approximation and demonstrates competitive performance on datasets like CIFAR10.
• The paper shows practical scalability and efficient sample generation, paving the way for real-time applications in image and audio synthesis.

Understanding Variational Schrödinger Diffusion Models

Introduction

Advances in diffusion models have significantly impacted various domains like image and audio synthesis. Traditional diffusion models, however, grapple with optimal transport properties that can make them inefficient for certain applications. A novel approach, leveraging the Schrödinger bridge problem, enhances these models by optimizing transport plans but at the cost of increased computational overhead due to intractable functions that necessitate simulated trajectories.

Addressing these concerns, the paper discusses the Variational Schrödinger Diffusion Model (VSDM), which simplifies these complexities through variational inference. This method linearizes the problematic functions, thereby making training more feasible without reliance on simulations.

Key Contributions of Variational Schrödinger Diffusion Models

The shift to a variational framework brings several notable enhancements and findings:

• Efficiency in Training: By approximating complex forward score functions with linear forms, VSDM reintroduces simulation-free properties in training, which boosts overall efficiency.
• Theoretical Robustness: Convergence of the variational scores is backed by stochastic approximation theories, ensuring that even with approximations, the system remains robust and converges effectively under certain conditions.
• Practical Scalability: The authors tested VSDM on complex data shapes and real-world datasets like CIFAR10, demonstrating competitive performance without the need for extensive tuning typically seen in large models.
• Straightforward Sample Trajectories: The VSDM encourages straighter and more efficient paths in the sample space, improving the quality of generation particularly in anisotropic data distributions.

Theoretical Implications

From a theoretical standpoint, VSDM introduces a balance by approximating certain components of the Schrödinger bridge problem, thus altering the traditional but computationally expensive approaches. This balance between computational feasibility and theoretical accuracy could pave the way for new research, especially in how variational methods can be applied to other complex models in machine learning.

Practical Implications

Practically, VSDM's ability to operate without extensive pre-computed simulations makes it a strong candidate for real-time applications or scenarios where computational resources are limited. Its performance on standard benchmarks like CIFAR10 illustrates its capability to handle complex, high-dimensional data efficiently, which is promising for applications in graphics generation, advanced simulations, and more.

Future Directions

The introduction of VSDM is a significant step, but the journey doesn't end here. The authors speculate that future developments might explore more dynamic approximations or even extend these techniques to other forms of differential equations used in modeling and simulation. There's also potential in exploring how different forms of variational inference can further optimize the trade-off between computational overhead and transport efficiency in diffusion models.

Conclusion

With VSDM, we witness a meaningful evolution in diffusion models, pushing the boundaries of efficiency and scalability while maintaining robust theoretical foundations. Its ability to generate quality data with reduced computational demands opens new avenues for both academic exploration and practical application in the field of AI and machine learning.