Your diffusion model secretly knows the dimension of the data manifold
This provides a new method for estimating data manifold dimensions, which is important for researchers in machine learning and data analysis, though it is incremental as it builds on existing diffusion model techniques.
The paper tackles the problem of estimating the intrinsic dimension of data manifolds by proposing a novel framework that leverages trained diffusion models, proving that the score function approximates the normal bundle and enabling dimension estimation. It outperforms established statistical estimators in controlled experiments on Euclidean and image data.
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.