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Vietoris-Rips Complexes of Split-Decomposable Spaces

Mario Gómez

TL;DR

This work advances topological data analysis of finite metrics by integrating split-decomposition theory with Vietoris–Rips persistent homology. It develops an explicit, efficiently verifiable framework for circular decomposable spaces, including an $O(n^2)$ recognition and a complete homotopy description for monotone metrics, grounded in the cyclic-graph VR analysis from prior work. For non-monotone circular decomposable spaces, it introduces a recursive Mayer–Vietoris scheme that expresses $H_*(\mathrm{VR}_r(X))$ in terms of a cyclic core and a non-cyclic remainder, enabling tractable homology computations. Beyond totally decomposable spaces, the paper exposes a block-decomposition of $\mathrm{VR}_r(X)$ via the tight span’s block structure, yielding a direct-sum decomposition of homology across blocks and offering practical speedups for PH computations in phylogenetics and related domains.

Abstract

Split-metric decompositions are an important tool in the theory of phylogenetics, particularly because of the link between the tight span and the class of totally decomposable spaces, a generalization of metric trees whose decomposition does not have a ``prime'' component. Their close relationship with trees makes totally decomposable spaces attractive in the search for spaces whose persistent homology can be computed efficiently. We study the subclass of circular decomposable spaces, finite metrics that resemble subsets of $\mathbb{S}^1$ and can be recognized in quadratic time. We give an $O(n^2)$ characterization of the circular decomposable spaces whose Vietoris-Rips complexes are cyclic for all distance parameters, and compute their homotopy type using well-known results on $\mathbb{S}^1$. We extend this result to a recursive formula that computes the homology of certain circular decomposable spaces that fail the previous characterization. Going beyond totally decomposable spaces, we identify an $O(n^3)$ decomposition of $\mathrm{VR}_r(X)$ in terms of the blocks of the tight span of $X$, and use it to induce a direct-sum decomposition of the homology of $\mathrm{VR}_r(X)$.

Vietoris-Rips Complexes of Split-Decomposable Spaces

TL;DR

This work advances topological data analysis of finite metrics by integrating split-decomposition theory with Vietoris–Rips persistent homology. It develops an explicit, efficiently verifiable framework for circular decomposable spaces, including an recognition and a complete homotopy description for monotone metrics, grounded in the cyclic-graph VR analysis from prior work. For non-monotone circular decomposable spaces, it introduces a recursive Mayer–Vietoris scheme that expresses in terms of a cyclic core and a non-cyclic remainder, enabling tractable homology computations. Beyond totally decomposable spaces, the paper exposes a block-decomposition of via the tight span’s block structure, yielding a direct-sum decomposition of homology across blocks and offering practical speedups for PH computations in phylogenetics and related domains.

Abstract

Split-metric decompositions are an important tool in the theory of phylogenetics, particularly because of the link between the tight span and the class of totally decomposable spaces, a generalization of metric trees whose decomposition does not have a ``prime'' component. Their close relationship with trees makes totally decomposable spaces attractive in the search for spaces whose persistent homology can be computed efficiently. We study the subclass of circular decomposable spaces, finite metrics that resemble subsets of and can be recognized in quadratic time. We give an characterization of the circular decomposable spaces whose Vietoris-Rips complexes are cyclic for all distance parameters, and compute their homotopy type using well-known results on . We extend this result to a recursive formula that computes the homology of certain circular decomposable spaces that fail the previous characterization. Going beyond totally decomposable spaces, we identify an decomposition of in terms of the blocks of the tight span of , and use it to induce a direct-sum decomposition of the homology of .
Paper Structure (30 sections, 49 theorems, 88 equations, 4 figures, 5 tables)

This paper contains 30 sections, 49 theorems, 88 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related work.
  3. Acknowledgments.
  4. Preliminaries
  5. Notation
  6. Background
  7. Split systems.
  8. Weakly compatible split systems.
  9. Split-metric decompositions.
  10. Circular collections of splits.
  11. Clique complexes of cyclic graphs.
  12. Buneman complex.
  13. Tight span.
  14. Tight spans of totally decomposable metrics.
  15. VR complexes of circular decomposable metrics
  16. ...and 15 more sections

Key Result

Theorem 2.5

Any metric $d_X:X \times X \to {\mathbb R}$ on a finite set $X$ decomposes as where $d_0$ is split-prime and the sum runs over all $d_X$-splits $A|B$. Moreover, the decomposition is unique in the following sense. Let ${\mathcal{S}}'$ be a weakly compatible split system on $X$ with weights $\lambda_S>0$ for $S \in {\mathcal{S}}'$. If $d_X = d_0 + \sum_{S \in {\mathcal{S}}'} \l

Figures (4)

  • Figure 1: Left: The complete bipartite graph $K_{2,3}$. There are no splits of $K_{2,3}$ into $d$-convex sets, so $K_{2,3}$ has no $d$-splits. Right: The hypercube graph $H_3$. The distance between any pair of white vertices is 2, so their $\beta_{\{a_1,a_2\}, \{b_1, b_2\}}$ coefficient is 0. This prevents $H_3$ from having any $d$-split.
  • Figure 2: The space $Y := \{x_1, y, x_3, x_4, x_5\}$ consists of the vertices of a regular pentagon inscribed on the circle. We attach an edge $e$ of length $\frac{1}{5}+\epsilon$ to the circle at the point $y$ and define $x_2$ as the boundary of $e$ different from $y$. The circular decomposition of $Y$ induces a circular decomposition of $X := \{x_1, x_2, x_3, x_4, x_5\}$ that satisfies $d_X(x_1,x_2), d_X(x_2, x_3) > d_X(x_1,x_3)$. See Example \ref{['ex:non-cyclic-5']}.
  • Figure 3: Given $12 < r <13$, the graph shown is the 1-skeleton of $\mathrm{VR}_r(X)$ where $X$ is the metric space whose distance matrix is shown in the right panel of Table \ref{['tab:recursive_space']}. $\mathrm{VR}_r(X)$ is not cyclic because it contains the edge $\{1,4\}$ despite not having all edges between the points $\{1,2,3,4\}$ or all edges between $\{4,5,6,7,1\}$. In particular, $E_X = \{ \{1,4\} \}$.
  • Figure 4: Runtime comparison of persistent homology computed using block decomposition (Corollary \ref{['cor:VR_wedge']} and BloDec) against Ripser. The input metrics have a varying number of blocks ($x$-axis), with each block being a uniformly sampled set of 20 points from 3-spheres of varying radii.

Theorems & Definitions (123)

  • Definition 2.1: Cyclic order
  • Definition 2.2: Circular sum
  • Remark 2.3
  • Example 2.4
  • Theorem 2.5: metric-decomposition
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8: Totally decomposable spaces
  • Theorem 2.9: metric-decomposition
  • Definition 2.10
  • ...and 113 more