Betti Curves of Rank One Symmetric Matrices

TL;DR

This paper characterizes Betti curves of rank one symmetric matrices using persistent homology and demonstrates their application in analyzing neural activity data from zebrafish calcium imaging.

Contribution

It provides a complete theoretical characterization of Betti curves for rank one symmetric matrices and illustrates their relevance in real biological data analysis.

Findings

Betti curves of rank one matrices are fully characterized by three theorems.

Betti curve signatures can be observed in neural activity data from zebrafish.

The approach aids in detecting underlying structure in biological datasets.

Abstract

Betti curves of symmetric matrices were introduced in (Giusti et. al., 2015) as a new class of matrix invariants that depend only on the relative ordering of matrix entries. These invariants are computed using persistent homology, and can be used to detect underlying structure in biological data that may otherwise be obscured by monotone nonlinearities. Here we prove three theorems that fully characterize the Betti curves of rank 1 symmetric matrices. We then illustrate how these Betti curve signatures arise in natural data obtained from calcium imaging of neural activity in zebrafish.