On the Asymptotic Mean Square Error Optimality of Diffusion Models

2024-03-05

AI Summary

Diffusion models (DMs) lack theoretical guarantees of mean squared error (MSE) optimality for denoising tasks. Method: We propose an efficient, deterministic denoising framework grounded in the structure of conditional mean estimators. Contributions/Results: First, we establish a non-asymptotic theoretical guarantee that DM denoisers polynomially converge to the MSE-optimal conditional mean estimator. Second, we reveal their dual nature: they simultaneously achieve asymptotic denoising optimality and encode strong generative priors. Third, we derive an explicit Lipschitz constant bound dependent solely on model hyperparameters. Our theory ensures rapid convergence under mild regularity conditions. Extensive experiments across multiple benchmark datasets empirically validate the framework’s efficiency and robustness in approximating the MSE-optimal solution.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Diffusion models (DMs) as generative priors have recently shown great potential for denoising tasks but lack theoretical understanding with respect to their mean square error (MSE) optimality. This paper proposes a novel denoising strategy inspired by the structure of the MSE-optimal conditional mean estimator (CME). The resulting DM-based denoiser can be conveniently employed using a pre-trained DM, being particularly fast by truncating reverse diffusion steps and not requiring stochastic re-sampling. We present a comprehensive (non-)asymptotic optimality analysis of the proposed diffusion-based denoiser, demonstrating polynomial-time convergence to the CME under mild conditions. Our analysis also derives a novel Lipschitz constant that depends solely on the DM's hyperparameters. Further, we offer a new perspective on DMs, showing that they inherently combine an asymptotically optimal denoiser with a powerful generator, modifiable by switching re-sampling in the reverse process on or off. The theoretical findings are thoroughly validated with experiments based on various benchmark datasets

Problem

• Analyzes MSE optimality of diffusion models
• Proposes novel denoising strategy inspired by CME
• Demonstrates polynomial-time convergence to CME

Innovation

• MSE-optimal conditional mean estimator
• Truncated reverse diffusion steps
• Novel Lipschitz constant derivation