Latent Manifold Dimension from Noisy Observations: Identifiability and Recovery via Diffusion Scores

Diffusion models trained by denoising score matching (DSM) offer a promising approach to intrinsic dimension (ID) estimation. However, when observations are not exactly supported on a manifold, two questions remain: **(Q1)** Does the observed law determine a unique latent dimension, and **(Q2)** which diffusion scales recover it? We address these questions under bounded normal tubular noise, allowing a broad range of noise distributions. For (Q1), we define *latent manifold dimension* (LMD) relative to a prescribed model class and establish a uniform reach-separation condition for identifiability that cannot generally be relaxed at its boundary. To study recovery, we first introduce *local Hessian spectral dimension* (LHSD) as a score-based dimension statistic and examine how accurately it can be estimated from data at a given diffusion scale. Under suitable regularity and thin-tube conditions, we establish its minimax estimation rate, attained by an unrestricted empirical-DSM reference estimator. We then address (Q2): LHSD can fail at fine scales and tends to zero as the diffusion scale grows, while a sharp approximation bound characterizes recovery at intermediate scales. Experiments with diffusion transformers qualitatively support this scale dependence, including accurate estimation of the latent dimension at intermediate scales.