  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.
