  Score-based generative models (SGMs) need to approximate the scores $\nabla
\log p_t$ of the intermediate distributions as well as the final distribution
$p_T$ of the forward process. The theoretical underpinnings of the effects of
these approximations are still lacking. We find precise conditions under which
SGMs are able to produce samples from an underlying (low-dimensional) data
manifold $\mathcal{M}$. This assures us that SGMs are able to generate the
"right kind of samples". For example, taking $\mathcal{M}$ to be the subset of
images of faces, we find conditions under which the SGM robustly produces an
image of a face, even though the relative frequencies of these images might not
accurately represent the true data generating distribution. Moreover, this
analysis is a first step towards understanding the generalization properties of
SGMs: Taking $\mathcal{M}$ to be the set of all training samples, our results
provide a precise description of when the SGM memorizes its training data.
