A Memorization–Generalization Phase Transition for Diffusion Scores

We study empirical diffusion scores for data supported on low-dimensional sets in $\mathbb R^D$. Our starting point is that, at noise level $t$, the Gaussian kernel probes neighborhoods of radius $\sqrt t$; on a $k$-dimensional support, this gives an effective local sample size of order $Nt^{k/2}$. We first consider Gaussian data supported on a $k$-dimensional linear subspace, where intrinsic and ambient dimensions can be separated exactly, and discuss a three-regime transition governed by $Nt^{k/2}$: single-sample domination when it vanishes, a nontrivial Poissonian random limit when it stays of order one, and convergence to the population score when it diverges. We then establish the analogous trichotomy for the uniform law on $\mathbb S^{D-1}$, with the corresponding parameter $Nt^{(D-1)/2}$. These two solvable geometries support $Nt^{k/2}$ as the natural intrinsic-dimension parameter for the memorization--generalization transition on smooth data manifolds.