Persistent homology of toroidal sliding window embeddings

Abstract

In this paper we study the persistent homology of sliding window embeddings of quasi-periodic signals. That is, functions which are sums of harmonics with different periods, and in particular, those having incommensurate frequencies. The resulting sliding window point-clouds, in this case, are (dense in) high-dimensional tori. We prove theorems which guide the choice of window size and embedding dimension, and describe the associated persistent homology.