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Identifying Topological Differences in Two Populations of Random Geometric Objects

Satish Kumar, Subhra Sankar Dhar

Abstract

We propose a statistical framework to identify topological differences in two populations of random geometric objects. The proposed framework involves first associating a topological signature with random geometric objects and then performing a two-sample test using the observed topological signatures. We associate persistence barcodes, a topological signature from topological data analysis, with each observed random geometric object. This, in turn, yields a two-sample problem on the space of persistence barcodes. As the space of persistence barcodes is not suitable for standard statistical analysis, we translate the two-sample problem on a suitable subset of a Euclidean space. In the course of this study, we embed the topological signatures in an ordered convex cone in a Euclidean space using functions from tropical geometry. We show that the embedding is a sufficient statistic for the persistence barcodes. This fact leads to the proposal of a two-sample test based on this sufficient statistic, and its equivalence to the two-sample problem on the barcode space is established. Finally, the consistency of the proposed test is studied.

Identifying Topological Differences in Two Populations of Random Geometric Objects

Abstract

We propose a statistical framework to identify topological differences in two populations of random geometric objects. The proposed framework involves first associating a topological signature with random geometric objects and then performing a two-sample test using the observed topological signatures. We associate persistence barcodes, a topological signature from topological data analysis, with each observed random geometric object. This, in turn, yields a two-sample problem on the space of persistence barcodes. As the space of persistence barcodes is not suitable for standard statistical analysis, we translate the two-sample problem on a suitable subset of a Euclidean space. In the course of this study, we embed the topological signatures in an ordered convex cone in a Euclidean space using functions from tropical geometry. We show that the embedding is a sufficient statistic for the persistence barcodes. This fact leads to the proposal of a two-sample test based on this sufficient statistic, and its equivalence to the two-sample problem on the barcode space is established. Finally, the consistency of the proposed test is studied.
Paper Structure (18 sections, 6 theorems, 60 equations)

This paper contains 18 sections, 6 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Random Geometric Objects
  3. Literature Review
  4. Our Contribution
  5. Mathematical Challenges
  6. Organization
  7. Preliminaries
  8. Barcode Space
  9. Tropical Functions
  10. Tropical Embedding
  11. Problem Formulation
  12. Equivalent Hypothesis Formulation
  13. Procedure: Test of Hypothesis
  14. Asymptotic Properties of Test
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 2.1

(Theorem 6.3 of Kališnik2019) Fix $n \in \mathbb{N}$, and for $i, j, k \in \{0, \ldots,n\}$ such that $(i + j + k) \leq n$, consider the family of functions $\left\{{T_m^{(i,j,k)} : m \in \mathbb{N}}\right\}$ on $\mathscr{B}_{\leq n}$, defined by Then, for two distinct point $\mathscr{B}_1$ and $\mathscr{B}_2$ in $\mathscr{B}_{\leq n}$, there exists $(i, j, k)$ such that $T_m^{(i,j,k)}(\mathscr{B

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.1
  • Theorem 2.1
  • Example 2.2
  • Theorem 3.1
  • Theorem 3.2
  • ...and 11 more