Combinatorial Persistent Homology Transform

TL;DR

This work reframes the persistent Homology Transform (PH transform) by interpreting the combinatorial persistence diagrams, obtained via Möbius inversion, as a functorial representation of a cellulation of the sphere ${\mathbb S}^N$ into the category of combinatorial persistence diagrams. By constructing a finite, essential cellulation from a geometric embedding, the authors define a functor from the cellulation to filtrations over a simplicial complex and, subsequently, to charge-preserving morphisms between combinatorial diagrams, yielding a finite, computable representation. They demonstrate the approach with concrete 2D and 3D examples, showing how filtrations and diagrams are consistent across cells and how the Möbius inversion recovers augmented persistence diagrams. The framework promises to leverage algebraic and topological combinatorics tools for analyzing parameterized shape descriptors and opens questions about global co-sections and stability across families of diagrams.

Abstract

The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on $\mathbb{S}^n$ to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes.

Figures (6)

• Figure 1: The V, embedded in ${\mathbb R}^2$. This simplicial complex has three vertices and two edges. By exploring the combinatorial PH transform for this example, we illustrate each step of the construction.
• Figure 2: In (\ref{['fig:ex-v-lines']}), we see the three linear subpaces of ${\mathbb R}^2$ that are used to define the cellulation over ${\mathbb S}^1$. In (\ref{['fig:ex-v-sphere']}), each cell is labeled by a vector in $\{-,0,+\}^3$ according to which side of $S_{1,2}$, $S_{1,3}$, and $S_{2,3}$ the cell falls. For example, the vector $(-++)$ labels the one-cell whose points are all in $S_{1,2}^-$, $S_{1,3}^+$, and $S_{2,3}^+$. The vector $(0++)$ labels the zero-cell that is in $S_{1,2}$, $S_{1,3}^+$, and $S_{2,3}^+$. In fact, $C_{(0++)}=S_{1,2}\cap S_{1,3}^+\cap S_{2,3}^+$. Note that no label is $(000)$, and that all labels are distinct. The partial order of the cells is denoted by arrows (where $a \to b$ indicates that $b < a$).
• Figure 3: The bounded lattice functions for the three highlighted face relations in Figure \ref{['fig:ex-v']}(\ref{['fig:ex-v-sphere']}). Notice that the two maps into ${{P}}_{(0++)}$ are nearly bijections, except for two vertices ($v_1$ and $v_2$) that map to the same equivalence class for both maps. This corresponds to the transposition of the two vertices between directions in $C_{(+++)}$ and $C_{(-++)}$. In fact, this property holds more generally for any face relation between codimension-one cells.
• Figure 4: The maps $\overline{P_{(+++)}} \to \overline{P_{0++}}$ and ${{\mathsf{ZB}}}_0(\overline{P_{(+++)}} \to \overline{P_{0++}})$. In both maps, the objects in the same pink region get mapped to the same object in the codomain. Using the field $\mathsf{k} = {\mathbb Z} / 2{\mathbb Z}$, for $b\neq \top$, ${{\mathsf{ZB}}}_0[a,b]$ counts the number of zero-cycles in the simplicial complex ${F}(a)$ that are zero-boundaries in the larger simplicial complex ${F}(b)$. ${{\mathsf{ZB}}}_0[a,\top]$ is a count of the number of vertices in ${F}(a)$.
• Figure 5: A geometric complex in ${\mathbb R}^3$ with the induced cellulation of ${\mathbb S}^2$. The cellulation depends only on the vetices of the complex. Since the vertices are in general position, each great circle on ${\mathbb S}^2$ is distinct.

Theorems & Definitions (8)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Proposition 2.6
• Proposition 3.1
• Definition 3.2