Distance Matrices of Ordered Point Clouds and Their Persistent Homology

What the math gives to ML

The paper constructs a filtration-compatible degree-one chain map from a two-dimensional recurrence or distance-matrix complex of an ordered trajectory into the Vietoris-Rips or Cech complex of its state-space samples. A recurrence-pixel vertex (i,j) is sent to the trajectory path from x_i to x_j closed by the shortcut edge [i,j], while grid edges and squares map to triangles and tetrahedra; explicit boundary identities guarantee that this is a genuine chain map. The transferable asset is a cheaper way to extract state-space cycling information from temporal embeddings without constructing a full high-dimensional Rips complex. In neural sequence models, this can become a topology-aware auxiliary loss or feature extractor based on persistence of recurrence plots, with representative grid cycles converted into geometric loops in the learned latent space.