A Faithful Discretization of the Verbose Persistent Homology Transform

TL;DR

This work provides a provable finite faithful discretization of the verbose PHT (VPHT) for arbitrary simplicial complexes in $\,\mathbb{R}^d$, achieving a discretization that is exponential in the dimension and stable under perturbations. The core idea is to enforce vertex- and simplex-isolating sets of directions and to reconstruct the original complex from the discretized VPHT via a reconstruction algorithm that uses filtration hyperplanes, $k$-indegree counts, and controlled tilts. The approach extends to related verbose transforms, including the VBFT and, with adaptations, the VECFT, yielding faithful discretizations of a broad class of dimension-returning descriptors. An explicit construction of the direction set is provided, with detailed auxiliary tools and a worked 3D example illustrating the methodology. The results pave the way for reliable, scalable discretizations suitable for machine learning and geometric inference while offering stability guarantees and explicit algorithms.

Abstract

The persistent homology transform (PHT) represents a shape with a multiset of persistence diagrams parameterized by the sphere of directions in the ambient space. In this work, we describe a finite set of diagrams that discretize the PHT such that it faithfully represents the underlying shape. We provide a discretization that is exponential in the dimension of the shape. Moreover, we show that this discretization is stable with respect to various perturbations and we provide an algorithm for computing the discretization. Our approach relies only on knowing the heights and dimensions of topological events, which means that it can be adapted to provide discretizations of other dimension-returning topological transforms, including the Betti function transform. With mild alterations, we also adapt our methods to faithfully discretize the Euler characteristic function transform.

Figures (6)

• Figure 1: The filtration hyperplanes (shaded in pink) corresponding to a pair of $[v_1, v_2, v_8]$-isolating directions, where we have $V=\{v_1, v_2, v_8\}$ and $W= \{v_8\}$. One hyperplane corresponds to a direction $s_V$ that is $K_0$-perpendicular to $V = [v_1, v_2, v_8]$. The other corresponds to a direction that is a $(K, V, W, s_V)$-perturbation) which, by pivoting $s_V$ around $V\setminus W$, "pops" the vertex of $W$ above the filtration hyperplane.
• Figure 2: The vertex set $P$ (large black points) defines the filtration grid of $P$ with respect to $\{e_1, e_2\}$, denoted $A$ (grey and black dots in the left figure). The direction $s$, indicated on the right, uniquely orders the points of $A$. Thus, since $e_1$ and $e_2$ are linearly independent and since the direction $s$ uniquely orders the points of the $A$, the set $\{e_1, e_2, s\}$ is vertex-isolating for the given vertex set. To locate the vertices of the set, we simply need to identify all intersections of $\mathbb{H}(s, P)$ (diagonal pink dashed lines) with $A$.
• Figure 3: Computing the three-indegree for a two-simplex (triangle) in $\mathbb{R}^4$. The simplex $\sigma$ is shown in dark gray. The direction $s \in \mathbb{S}^3$ is orthogonal to $\mathop{\mathrm{aff}}\limits(\sigma)$ such that all other vertices shown are below $\sigma$ (note that $s=s_{\sigma}$ from Definition \ref{['def:simplexiso']}(\ref{['stmt:prop:simplexiso-perp']})). Although the three-indegree of $\sigma$ is one, the VPD in direction $s$ sees three tetrahedron at the same height as $\sigma$. Recursively using the three-indegree of all faces of $\sigma$ in tilted directions (that is, directions given in Definition \ref{['def:simplexiso']}(\ref{['stmt:prop:simplexiso-tiltdown']})), the three-indegree of $\sigma$ can be defined by subtracting the indegrees of faces of $\sigma$ in specific directions ($3-1-1=1$) given in Equation (\ref{['eqn:computekindegree']}).
• Figure 4: The solid grey lines in the figure above indicate the changing heights of points as we swing direction $s$ towards $s'$. Although we do not explicitly compute the grey lines, we know by simple geometry that no intersection of grey lines (and, in particular, no swapping of point orders) occurs before the $t$-value $t_*$, which corresponds to the intersection of the closest pairwise heights of points on the left and the extremal heights of points on the right, as indicated by the black lines. Since there are no crossings of line segments before $\frac{1}{2}t_*$, there is therefore no change in the order of points with respect to direction $s_t = (1-\frac{1}{2}t_*)s + \frac{1}{2}t_* s'$.
• Figure 5: A two-simplex in $\mathbb{R}^3$ (left) stratifies $\mathbb{S}^2$ where each stratum is a region containing all directions that define the same partial order on vertices. Notice that the set of directions perpendicular to any pair of vertices forms a great circle and the two directions perpendicular to $[v_1,v_2,v_3]$ correspond to the two three-way intersections of these great circles.

Theorems & Definitions (75)

• Definition 0: Verbose Persistence Diagram
• Definition 1: Verbose Persistent Homology Transform
• Definition 2: Faithful Discretization of $VPHT(K)$
• Definition 3: Betti Function (BF) and Verbose BF
• Corollary 4: Properties of VBFs
• Definition 5: Verbose Betti Function Transform
• Definition 6: Euler Characteristic Function (ECF) and Verbose ECF
• Remark 7: From Persistence Diagrams to ECFs
• Corollary 8: Properties of VECFs
• Definition 9: Verbose Euler Characteristic Function Transform