  Prompting has become an important mechanism by which users can more
effectively interact with many flavors of foundation model. Indeed, the last
several years have shown that well-honed prompts can sometimes unlock emergent
capabilities within such models. While there has been a substantial amount of
empirical exploration of prompting within the community, relatively few works
have studied prompting at a mathematical level. In this work we aim to take a
first step towards understanding basic geometric properties induced by prompts
in Stable Diffusion, focusing on the intrinsic dimension of internal
representations within the model. We find that choice of prompt has a
substantial impact on the intrinsic dimension of representations at both layers
of the model which we explored, but that the nature of this impact depends on
the layer being considered. For example, in certain bottleneck layers of the
model, intrinsic dimension of representations is correlated with prompt
perplexity (measured using a surrogate model), while this correlation is not
apparent in the latent layers. Our evidence suggests that intrinsic dimension
could be a useful tool for future studies of the impact of different prompts on
text-to-image models.
