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.

翻译：在这项工作中，我们提出了一种利用训练好的扩散模型估计数据流形维数的新框架。扩散模型近似于得分函数，即目标分布在不同噪声水平下的噪声扰动版本的密度对数梯度。我们证明，若数据集中在嵌入高维空间中的流形附近，则随着噪声水平降低，得分函数指向流形方向，因为该方向成为似然增长最大的方向。因此，对于较小的噪声水平，扩散模型使我们能够近似获取数据流形的法丛。这使我们能够估计切空间的维数，从而得到数据流形的内在维数。据我们所知，我们的方法是首个基于扩散模型的数据流形维数估计器，并且在欧几里得数据和图像数据的受控实验中，其性能优于成熟的统计估计器。