Probing the topology of the space of tokens with structured prompts

The set of tokens T, when embedded within the latent space X of a large language model (LLM) can be thought of as a finite sample drawn from a distribution supported on a topological subspace of X. One can ask what the smallest (in the sense of inclusion) subspace and simplest (in terms of fewest free parameters) distribution can account for such a sample. Previous work[1] suggests that the smallest topological subspace from which tokens can be drawn is not manifold, but has structure consistent with a stratified manifold. That paper relied upon knowing the token input embedding function T X, which given each token t T, ascribes a representation in X. Because embeddings preserve topological structure, in this paper, we will study T by equating it with the image of the token input embedding function, thereby treating T both as the set of tokens and as a subspace of X. This subspace is called the token subspace of X. Usually X is taken to be Euclidean space R