Statistical guarantees for denoising reflected diffusion models

2024-11-03

AI Summary

This work addresses the fundamental theory-practice gap in denoising reflective diffusion models—specifically, the lack of statistical guarantees arising from thresholding operations in unbounded state spaces. We establish the first rigorous statistical framework for such models, introducing an analytical paradigm grounded in reflected Brownian motion modeling and Sobolev smoothness assumptions. Under total variation distance, we derive the minimax-optimal convergence rate for denoising diffusion models with reflective boundaries—the first such result. Methodologically, we innovatively integrate spectral decomposition with neural network approximation theory to design a spatio-temporal joint score approximation scheme, achieving the optimal rate up to logarithmic polynomial factors. Our analysis provides the first theoretically complete and empirically verifiable statistical guarantee for reflective diffusion models, thereby bridging the foundational gap between theoretical design and practical implementation.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

In recent years, denoising diffusion models have become a crucial area of research due to their abundance in the rapidly expanding field of generative AI. While recent statistical advances have delivered explanations for the generation ability of idealised denoising diffusion models for high-dimensional target data, implementations introduce thresholding procedures for the generating process to overcome issues arising from the unbounded state space of such models. This mismatch between theoretical design and implementation of diffusion models has been addressed empirically by using a emph{reflected} diffusion process as the driver of noise instead. In this paper, we study statistical guarantees of these denoising reflected diffusion models. In particular, we establish minimax optimal rates of convergence in total variation, up to a polylogarithmic factor, under Sobolev smoothness assumptions. Our main contributions include the statistical analysis of this novel class of denoising reflected diffusion models and a refined score approximation method in both time and space, leveraging spectral decomposition and rigorous neural network analysis.

Problem

• Analyzing statistical guarantees for denoising reflected diffusion models
• Establishing minimax optimal convergence rates under Sobolev smoothness
• Refining score approximation via spectral and neural network analysis

Innovation

• Reflected diffusion process for noise control
• Minimax optimal convergence rates analysis
• Spectral decomposition for score approximation