Persistent Cycle Representatives and Generalized Landscapes for Codimension 1 Persistent Homology

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.

💡 Research Summary

The paper introduces a novel framework for extracting geometrically meaningful representatives of homology classes in codimension‑one persistent homology, i.e., for the d‑dimensional homology of a (d + 1)‑dimensional simplicial complex K embedded in ℝ^{d+1}. The key insight is to use Alexander duality: the complement ℝ^{d+1}\K can be triangulated as a dual cell complex \bar{K}, and the connected components of \bar{K} correspond bijectively to volume‑optimal d‑cycles in K. By assigning a weight function to (d + 1)‑simplices (typically the constant weight 1) the authors prove that the set of cycles obtained from the connected components forms a minimal (and, when all weights are positive, unique) cycle basis for H_d(K).

The authors extend this correspondence to filtrations. For a filtered complex K = {K_t}_{t∈ℝ}, the dual filtration \bar{K}t = K{-t} yields a merge tree (or, after reduction, a merge forest) that records how the connected components of the complement appear, merge, and disappear as t varies. Each node of the forest represents a component at a specific time, and each edge corresponds to a merge event. This structure induces a “persistence forest” that aligns perfectly with the barcode of H_d(K): every bar I =