Your diffusion model secretly knows the dimension of the data manifold

TL;DR

This paper introduces a novel method leveraging diffusion models to estimate the intrinsic dimension of data manifolds, outperforming existing estimators in experiments on Euclidean and image data.

Contribution

It presents the first diffusion-model-based estimator for data manifold dimension, connecting score functions to manifold geometry for the first time.

Findings

Outperforms traditional estimators in controlled experiments

Works effectively on both Euclidean and image data

Provides a new geometric perspective using diffusion models

Abstract

In this work, we propose a novel framework for estimating the dimension of the data manifold using a trained diffusion model. A diffusion model approximates the score function i.e. the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that, if the data concentrates around a manifold embedded in the high-dimensional ambient space, then as the level of corruption decreases, the score function points towards the manifold, as this direction becomes the direction of maximal likelihood increase. Therefore, for small levels of corruption, the diffusion model provides us with access to an approximation of the normal bundle of the data manifold. This allows us to estimate the dimension of the tangent space, thus, the intrinsic dimension of the data manifold. To the best of our knowledge, our method is the first estimator of the data manifold dimension based on diffusion models and it outperforms well established statistical estimators in controlled experiments on both Euclidean and image data.