Topological simplification guided by forbidden regions

Abstract

Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c$\cdot$n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.

Figures (5)

• Figure 1: Two vector fields differing by a reversal of the path between components of a birth-death pair $\alpha$. Critical cells are shown with colored nodes, and arrows between them symbolize paths created by vectors. Above each vector field is the boundary matrix of the corresponding Morse complex. Reversing the path between components of $\alpha$ gives the same boundary matrix as performing the Lefschetz cancelation.
• Figure 2: Left: $(n-1)$-st dimensional persistence diagram of some complex $X$, with $D_{n}$ in the bottom-right corner. In the diagram, we denote by $\times$ homological relations between pairs, and by $\otimes$ relations which are homological and cohomological at the same time. To decide if moving $\beta^\circ$ past $\alpha^\circ$ changes the relations between cells, as determined by \ref{['eq:death_death_update']} in \ref{['thm:birth_transposition']}, we need to calculate $D_{n}^{\alpha,\beta}$. Right: The persistence diagram with the updated relation after the transposition of $\alpha^\circ$ and $\beta^\circ$. In the bottom-right corner, we show $D_{n}^{\alpha,\beta}$ before the transposition. The cells deleted by the Lefschetz cancellations are crossed out.
• Figure 3: Top: Schematic picture of $\mathcal{V}^{k}_{h}$. Node heights encode values of $\text{dMf}$, critical cells are labeled by Greek letters with superscripts. Several important sublevels are highlighted with dashed lines. Bottom: Boundary matrix and the persistence diagram of the Morse complex induced by $\mathcal{V}^{k}_{h}$ with the two kinds of forbidden regions highlighted, and relations involving the birth-death pair $\alpha$ shown as edges. The forbidden regions for $\alpha^{\!\times}$ are shown in light blue; those for $\alpha^\circ$, in darker blue. The dashed arrows illustrate a possible homotopy, which moves the point to the diagonal.
• Figure 4: Persistence diagram of the 10-simplex, filtered by a random injective $\text{dMf}$ such that every birth-death pair of dimension $n$ is separated from pairs of dimensions $n+1$ and $n-1$. We apply a procedure that first simplifies $\text{dMf}$ by the standard method, i.e., path reversing between shallow pairs. When there is no reversible shallow pair left, we continue using the algorithm described in this paper. We made multiple passes canceling any pair that met the algorithm’s assumptions. We stopped when there was no reversible pair with a path between forbidden regions. Pairs of different types (canceled by the standard method, canceled using forbidden regions, not cancelable) are denoted by different colors. Figure \ref{['fig:result_zoom']} zooms-in on the pairs in dimension 4.
• Figure 5: Birth-death pairs in dimension 4 from Figure \ref{['fig:result']}.

Theorems & Definitions (61)

• Theorem 1
• Definition 2
• Definition 3
• Definition 5
• Definition 6
• Definition 7: Morse complex
• Corollary 8
• Theorem 9: Forman2002
• Definition 10
• Definition 11