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Inferring Ambient Cycles of Point Samples on Manifolds with Universal Coverings

Ka Man Yim

TL;DR

This work develops a groupoid-based framework for inferring ambient topological features of a finite point cloud in a compact Riemannian manifold by exploiting the universal covering. It connects edge-path data, monodromy, and transition maps on nerves of good covers to recover induced maps on ${\pi_1}$ and ${H_1}$, with practical pipelines using either a thickened Čech approach (for $\,\epsilon<\mathrm{conv}(M)$) or min-geodesic graphs when sampling is sparse. The key contributions include formalizing the equivalence between edge groupoids and fundamental groupoids, deriving a finite transition data method to sample monodromy, and applying this to decompose principal persistence measures of four-point cycles by ambient homology classes on model manifolds. The results provide a tangible route to interpret geometric signatures in persistence diagrams through ambient topology, enabling more robust topological data analysis on manifolds.

Abstract

A central objective of topological data analysis is to identify topologically significant features in data represented as a finite point cloud. We consider the setting where the ambient space of the point sample is a compact Riemannian manifold. Given a simplicial complex constructed on the point set, we can relate the first homology of the complex with that of the ambient manifold by matching edges in the complex with minimising geodesics between points. Provided the universal covering of the manifold is known, we give a constructive method for identifying whether a given edge loop (or representative first homology cycle) on the complex corresponds to a non-trivial loop (or first homology class) of the ambient manifold. We show that metric data on the point cloud and its fibre in the covering suffices for the construction, and formalise our approach in the framework of groupoids and monodromy of coverings.

Inferring Ambient Cycles of Point Samples on Manifolds with Universal Coverings

TL;DR

This work develops a groupoid-based framework for inferring ambient topological features of a finite point cloud in a compact Riemannian manifold by exploiting the universal covering. It connects edge-path data, monodromy, and transition maps on nerves of good covers to recover induced maps on and , with practical pipelines using either a thickened Čech approach (for ) or min-geodesic graphs when sampling is sparse. The key contributions include formalizing the equivalence between edge groupoids and fundamental groupoids, deriving a finite transition data method to sample monodromy, and applying this to decompose principal persistence measures of four-point cycles by ambient homology classes on model manifolds. The results provide a tangible route to interpret geometric signatures in persistence diagrams through ambient topology, enabling more robust topological data analysis on manifolds.

Abstract

A central objective of topological data analysis is to identify topologically significant features in data represented as a finite point cloud. We consider the setting where the ambient space of the point sample is a compact Riemannian manifold. Given a simplicial complex constructed on the point set, we can relate the first homology of the complex with that of the ambient manifold by matching edges in the complex with minimising geodesics between points. Provided the universal covering of the manifold is known, we give a constructive method for identifying whether a given edge loop (or representative first homology cycle) on the complex corresponds to a non-trivial loop (or first homology class) of the ambient manifold. We show that metric data on the point cloud and its fibre in the covering suffices for the construction, and formalise our approach in the framework of groupoids and monodromy of coverings.

Paper Structure

This paper contains 21 sections, 47 theorems, 86 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Coverings and Groupoids
  3. Applications to Topological Data Analysis on Manifolds
  4. Related Work
  5. Groupoids
  6. The Fundamental Groupoid and the Edge Groupoid
  7. Equivalence of the Fundamental and Edge Groupoids
  8. Homology
  9. Covering Spaces
  10. Monodromy
  11. Transition Maps of the Covering
  12. Induced Coverings
  13. Recovering Induced Coverings on Data from Geodesics
  14. Geometry Preliminaries
  15. Inference of Ambient Cycles
  16. ...and 6 more sections

Key Result

Lemma 2.8

Let $K$ be a simplicial complex and $\iota: K_1 \hookrightarrow K$ be the inclusion of its one-skeleton into $K$. Suppose we have a groupoid homomorphism $F_1: \mathcal{E} {{K_1}} \to \mathsf{C}$. Then there is a groupoid homomorphism $F: \mathcal{E} {{K}} \to \mathsf{C}$ such that $F_1 = F \circ \m

Figures (5)

  • Figure 1: Consider the boundary of a two-simplex $K$ and the three possible surjective simplicial maps from $K$ to the closure of edge. The images of the maps on vertices is marked out in the diagram. For the (arbitrarily chosen) snapping on the left hand side edge, we illustrate the only snapping on the bottom edge, such that there is a snapping on $K$ that is simultaneously compatible with snappings on both the left and bottom edge. There is no snapping on the remaining edge that is compatible with this snapping. In other words, there is no choice of snappings on the three codomains, where there is a snapping on the pre-image such that \ref{['dgm:snapping_functorial']} commutes for all three maps.
  • Figure 2: Empirical samples of the first principal Vietoris-Rips persistence measure on the flat torus. The homology classes of the four point cycles are represented by $(n,m) \in \mathop{\mathrm{\mathbb{Z}}}\nolimits \oplus \mathop{\mathrm{\mathbb{Z}}}\nolimits \cong {H_{1}}\mathopen{}\left({M}\right)\mathclose{}$. We group those $(1,0)$ and $(0,1)$ into one diagram $(1,0)$ due to the symmetry between the generators.
  • Figure 3: Empirical samples of the first principal Vietoris-Rips persistence measure on the flat Klein bottle. The homology classes of the four point cycles are represented by $(n,m) \in \mathop{\mathrm{\mathbb{Z}}}\nolimits \oplus \mathop{\mathrm{\mathbb{Z}}}\nolimits_2 \cong {H_{1}}\mathopen{}\left({M}\right)\mathclose{}$.
  • Figure 4: Empirical samples of the first principal Vietoris-Rips persistence measure on $\mathop{\mathrm{\mathbb{R}P}}\nolimits^2$. The homology classes of the four point cycles are represented by $a \in \mathop{\mathrm{\mathbb{Z}}}\nolimits_2 \cong {H_{1}}\mathopen{}\left({M}\right)\mathclose{}$.
  • Figure 5: Empirical samples of the first principal Vietoris-Rips persistence measure on the orientable surface with genus two. The homology classes of the four point cycles are represented by $(n_0,\ldots, n_3) \in \mathop{\mathrm{\mathbb{Z}}}\nolimits^4 \cong {H_{1}}\mathopen{}\left({M}\right)\mathclose{}$. Due to symmetries betwen the the four generators, the homology class indicated in the diagrams represent classes up to permutation of $(n_0,\ldots, n_3)$.

Theorems & Definitions (100)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Example 2.9
  • ...and 90 more