Generalization error bound for denoising score matching under relaxed manifold assumption

2025-02-19

AI Summary

This paper investigates non-asymptotic error bounds for denoising score matching estimation under a relaxed manifold assumption—allowing data to deviate slightly from a low-dimensional manifold. We consider observations modeled by a nonparametric Gaussian mixture and propose the first theoretical framework that characterizes convergence rates in terms of intrinsic dimension rather than ambient dimension, under this manifold-relaxed setting where samples may depart from the manifold. Our main contributions are threefold: (1) we derive tight non-asymptotic upper bounds on both approximation and generalization errors; (2) we establish that these bounds remain valid even when the ambient dimension grows polynomially with the sample size; and (3) by relaxing the strict manifold assumption, we significantly enhance the theoretical applicability and practical relevance of score-based generative modeling for high-dimensional sparse and near-manifold data.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We examine theoretical properties of the denoising score matching estimate. We model the density of observations with a nonparametric Gaussian mixture. We significantly relax the standard manifold assumption allowing the samples step away from the manifold. At the same time, we are still able to leverage a nice distribution structure. We derive non-asymptotic bounds on the approximation and generalization errors of the denoising score matching estimate. The rates of convergence are determined by the intrinsic dimension. Furthermore, our bounds remain valid even if we allow the ambient dimension grow polynomially with the sample size.

Problem

• Relaxed manifold assumption analysis
• Nonparametric Gaussian mixture modeling
• Generalization error bounds derivation

Innovation

• Nonparametric Gaussian mixture model
• Relaxed manifold assumption
• Non-asymptotic error bounds