Schödinger Bridge Type Diffusion Models as an Extension of Variational Autoencoders (2412.18237v1)

Abstract: Generative diffusion models use time-forward and backward stochastic differential equations to connect the data and prior distributions. While conventional diffusion models (e.g., score-based models) only learn the backward process, more flexible frameworks have been proposed to also learn the forward process by employing the Schr\"odinger bridge (SB). However, due to the complexity of the mathematical structure behind SB-type models, we can not easily give an intuitive understanding of their objective function. In this work, we propose a unified framework to construct diffusion models by reinterpreting the SB-type models as an extension of variational autoencoders. In this context, the data processing inequality plays a crucial role. As a result, we find that the objective function consists of the prior loss and drift matching parts.

• The paper proposes a novel framework unifying Schrödinger Bridge diffusion models and Variational Autoencoders by extending VAEs with continuous latent paths.
• This framework enables training diffusion models by leveraging the data processing inequality, bypassing the need for complex score matching computations.
• Integrating the VAE perspective simplifies the interpretation and training of SB-type models while expanding the application range of VAEs.

Overview of "Schrödinger Bridge Type Diffusion Models as an Extension of Variational Autoencoders"

The paper "Schrödinger Bridge Type Diffusion Models as an Extension of Variational Autoencoders" presents a novel conceptual framework that unifies diffusion models with variational autoencoders (VAEs). This new representation offers a reinterpretation of Schrödinger Bridge (SB)-type models, which are augmented through the integration of VAEs' principles. The paper explores the construction and training of diffusion models by leveraging the data processing inequality, framing it within a VAE-inspired architecture.

Key Contributions

1. Unification Framework: The authors propose a framework that reinterprets SB-type diffusion models as an extension of VAEs. The core idea is to view diffusion models through the lens of VAEs by extending the number of latent variables to form paths, which allows for the application of the data processing inequality.
2. Generative Process and Objective Function: Within this framework, the diffusion model's generative process is explained akin to the VAE's encoder-decoder mechanism. The objective function consists of two terms: one for minimizing prior loss (akin to VAEs' Kullback-Leibler divergence between encoder outputs and prior) and another for drift matching between the forward (encode) and backward (decode) stochastic differential equations (SDEs).
3. Training Diffusion Models: The established framework facilitates the derivation of the diffusion models' training objectives without relying on complex computations such as explicit score matching, which is a limitation in conventional score-based models (SBMs).

Theoretical Implications

1. Reduction of Complexity: By integrating the VAE framework, the authors simplify the interpretation and training of SB-type models, which traditionally require the intricate handling of forward-backward SDEs and the nonlinear Feynman-Kac formula.
2. Expanding on VAEs: This approach showcases how VAEs can be extended beyond their traditional scope by integrating continuous latent paths, enhancing their flexibility and application range.
3. Closing the Gap with Schrödinger Bridge Problem: The paper implicitly solves the SB problem within the diffusion model framework; this alignment not only captures optimal transport dynamics but also addresses the prior mismatch issue in finite time horizons.

Practical Implications

1. Flexibility in Model Design: By allowing more complex SDE structures, models built within this framework can potentially generate data more accurately and efficiently, catering to applications that require high-quality synthesis, such as in image and audio processing.
2. Enhanced Sampling Methods: The introduction of a probabilistic-flow ODE as an alternative to SDEs for sample generation increases sampling speed and stability, although it requires precise model training for optimal performance.
3. Facilitating Model Implementation: The framework enables practical training without approximations typical of SBMs, which can potentially reduce computational overhead and improve model robustness.

Speculation on Future Developments

Given the trajectory of integration between VAE concepts and diffusion models, future research might explore further extensions such as hybrid models that incorporate elements from other generative paradigms like GANs or integrating adaptive neural architecture for more efficient score function evaluations. Additionally, there could be a focus on applications in areas requiring continuous data transformations, such as time-series analysis or geospatial data interpolations, underpinned by robust theoretical formulations like those proposed in this paper.

Ultimately, this paper contributes significant theoretical advancements to generative modeling by bridging critical gaps between VAEs and diffusion frameworks, supplying enhanced avenues for both academic exploration and practical implementation.