Variational Schrödinger Momentum Diffusion (2501.16675v1)

Abstract: The momentum Schr\"odinger Bridge (mSB) has emerged as a leading method for accelerating generative diffusion processes and reducing transport costs. However, the lack of simulation-free properties inevitably results in high training costs and affects scalability. To obtain a trade-off between transport properties and scalability, we introduce variational Schr\"odinger momentum diffusion (VSMD), which employs linearized forward score functions (variational scores) to eliminate the dependence on simulated forward trajectories. Our approach leverages a multivariate diffusion process with adaptively transport-optimized variational scores. Additionally, we apply a critical-damping transform to stabilize training by removing the need for score estimations for both velocity and samples. Theoretically, we prove the convergence of samples generated with optimal variational scores and momentum diffusion. Empirical results demonstrate that VSMD efficiently generates anisotropic shapes while maintaining transport efficacy, outperforming overdamped alternatives, and avoiding complex denoising processes. Our approach also scales effectively to real-world data, achieving competitive results in time series and image generation.

• The paper introduces Variational Schr ödinger Momentum Diffusion (VSMD), a novel generative model that leverages variational inference and kinetic diffusions to improve scalability and transport efficiency over previous methods like Momentum Schr ödinger Bridge.
• VSMD features simulation-free properties via variational scores and employs an adaptive transport-optimized diffusion process utilizing velocity variables, enabling efficient generation of anisotropic shapes and complex data.
• Empirical assessments demonstrate VSMD's robust performance across synthetic, image, and time series datasets, showcasing its ability to balance computational efficiency with effective handling of high-dimensional and anisotropic distributions.

Variational Schrödinger Momentum Diffusion

The paper introduces a novel generative model termed Variational Schrödinger Momentum Diffusion (VSMD), addressing the limitations and inefficiencies of existing diffusion models in generative modeling. The Momentum Schrödinger Bridge (mSB) serves as an antecedent method but suffers from high computational costs due to its dependency on simulated forward trajectories. VSMD is proposed to optimize the trade-off between scalability and transport properties via variational inference methodologies and kinetic diffusions.

Core Contributions

The research presents VSMD, which leverages variational inference, a critical-damping transform, and multivariate kinetic Langevin dynamics to improve upon the mSB framework. Key contributions are:

1. Simulation-Free Properties: VSMD employs variational scores that linearize forward score functions, minimizing the dependence on simulation. This process reduces computational expenses and enhances the scalability of the generative model.
2. Adaptive Diffusion: The model integrates an adaptively transport-optimized diffusion process, allowing it to efficiently generate data, especially anisotropic shapes, and outperform simpler models by leveraging additional velocity variables that aid in maintaining transport efficacy.
3. Theoretical Guarantees: A convergence proof is provided for sample quality under optimal variational scores, asserting that VSMD approaches the desired distribution more efficiently compared to other existing methods.

Empirical Results

The empirical assessments demonstrate that VSMD performs robustly across both synthetic datasets, such as anisotropically transformed spiral and checkerboard data, and real-world datasets in image and time series domains. Specifically:

• VSMD’s underdamped variant (VSULD) outperformed critically damped and other traditional models such as CLD and VSDM in generating shapes with varying degrees of anisotropy.
• In time series forecasting, VSMD variants achieved competitive performance metrics, indicating superior adaptability to multivariate temporal datasets.
• Although not the top-performing model in image quality as per FID scores, VSMD establishes itself as a scalable alternative capable of handling high-dimensional distributions efficiently.

Implications and Future Work

VSMD’s development marks a shift towards more computationally viable generative modeling strategies by leveraging transport optimization within a variational inference framework. This can significantly impact the fields of deep generative models and optimal transport theory, enabling more complex and high-dimensional data operations.

However, the model leaves room for enhancements, particularly concerning the exploration of preconditioning techniques and optimization of damping parameters to potentially boost its generative performance to match or exceed state-of-the-art models. Future research could explore extending this framework to accommodate more complex boundary conditions or additional constraints, enhancing its applicability in constrained generative scenarios.

In summary, the Variational Schrödinger Momentum Diffusion model represents an innovative approach to generative modeling, providing an effective balance between scalability, accuracy, and computational efficiency through its refined use of variational methods and momentum-based dynamics.