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In this work, we propose a novel framework for estimating the dimension ofthe data manifold using a trained diffusion model. A diffusion modelapproximates the score function i.e. the gradient of the log density of anoise-corrupted version of the target distribution for varying levels ofcorruption. We prove that, if the data concentrates around a manifold embeddedin the high-dimensional ambient space, then as the level of corruptiondecreases, the score function points towards the manifold, as this directionbecomes the direction of maximal likelihood increase. Therefore, for smalllevels of corruption, the diffusion model provides us with access to anapproximation of the normal bundle of the data manifold. This allows us toestimate the dimension of the tangent space, thus, the intrinsic dimension ofthe data manifold. To the best of our knowledge, our method is the firstestimator of the data manifold dimension based on diffusion models and itoutperforms well established statistical estimators in controlled experimentson both Euclidean and image data.

Your diffusion model secretly knows the dimension of the data manifold