Persistent Cycle Representatives and Generalized Landscapes for Codimension 1 Persistent Homology

TL;DR

This paper introduces a method to compute volume-optimal representative cycles for persistent homology in filtered complexes, enabling the creation of generalized landscapes that capture geometric shape information beyond traditional persistence diagrams.

Contribution

It develops an efficient algorithm for tracking representative cycles through filtrations and introduces generalized persistence landscapes that incorporate geometric functionals.

Findings

Efficient algorithm with quadratic time complexity for cycle progression.

Generalized landscapes distinguish shapes with similar persistent homology.

Standard persistence landscapes are recovered as a special case.

Abstract

For a filtered simplicial complex $K$ embedded in $\mathbb{R}^{d+1}$, the merge tree of the complement of $K$ induces a forest structure on the persistent homology $H_d(K)$ via Alexander duality. We prove that the connected components of $\mathbb{R}^{d+1}\setminus K_r$ correspond to representative cycles for a basis of $H_d(K_r)$ which are volume-optimal. By keeping track of how these representatives evolve with the filtration of $K$, we can equip each interval $I$ in the barcode of $H_d(K)$ with a sequence of canonical representative cycles. We develop and implement an efficient algorithm to compute the progression of cycles in time $\mathcal{O}((\#K)^2)$. We apply functionals to these representatives, such as path length, enclosed volume, or total curvature. This way, we obtain a real-valued function for each interval, which captures geometric information about~$K$. Deriving from this construction, we introduce the \emph{generalized persistence landscapes}. Using the constant one-function as the functional, this construction gives back the standard persistence landscapes. Generalized landscapes can distinguish point clouds with similar persistent homology but distinct shape, which we demonstrate by concrete examples.