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The Density of Cross-Persistence Diagrams and Its Applications

Alexander Mironenko, Evgeny. Burnaev, Serguei Barannikov

TL;DR

This work presents the first systematic study of the density of cross-persistence diagrams, proves its existence, establishes theoretical foundations for its statistical use, and designs the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices.

Abstract

Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales. While effective for analyzing individual manifolds, persistence diagrams do not account for interactions between pairs of them. Cross-persistence diagrams (cross-barcodes), introduced recently, address this limitation by characterizing relationships between topological features of two point clouds. In this work, we present the first systematic study of the density of cross-persistence diagrams. We prove its existence, establish theoretical foundations for its statistical use, and design the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices. Our statistical approach enables the distinction of point clouds sampled from different manifolds by leveraging the linear characteristics of cross-persistence diagrams. Interestingly, we find that introducing noise can enhance our ability to distinguish point clouds, uncovering its novel utility in TDA applications. We demonstrate the effectiveness of our methods through experiments on diverse datasets, where our approach consistently outperforms existing techniques in density prediction and achieves superior results in point cloud distinction tasks. Our findings contribute to a broader understanding of cross-persistence diagrams and open new avenues for their application in data analysis, including potential insights into time-series domain tasks and the geometry of AI-generated texts. Our code is publicly available at https://github.com/Verdangeta/TDA_experiments

The Density of Cross-Persistence Diagrams and Its Applications

TL;DR

This work presents the first systematic study of the density of cross-persistence diagrams, proves its existence, establishes theoretical foundations for its statistical use, and designs the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices.

Abstract

Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales. While effective for analyzing individual manifolds, persistence diagrams do not account for interactions between pairs of them. Cross-persistence diagrams (cross-barcodes), introduced recently, address this limitation by characterizing relationships between topological features of two point clouds. In this work, we present the first systematic study of the density of cross-persistence diagrams. We prove its existence, establish theoretical foundations for its statistical use, and design the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices. Our statistical approach enables the distinction of point clouds sampled from different manifolds by leveraging the linear characteristics of cross-persistence diagrams. Interestingly, we find that introducing noise can enhance our ability to distinguish point clouds, uncovering its novel utility in TDA applications. We demonstrate the effectiveness of our methods through experiments on diverse datasets, where our approach consistently outperforms existing techniques in density prediction and achieves superior results in point cloud distinction tasks. Our findings contribute to a broader understanding of cross-persistence diagrams and open new avenues for their application in data analysis, including potential insights into time-series domain tasks and the geometry of AI-generated texts. Our code is publicly available at https://github.com/Verdangeta/TDA_experiments
Paper Structure (32 sections, 10 theorems, 33 equations, 20 figures, 5 tables)

This paper contains 32 sections, 10 theorems, 33 equations, 20 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Background
  4. Persistence Barcodes
  5. Persistence Diagrams
  6. The density of expected persistence diagrams
  7. Cross-Barcode (P, Q)
  8. The density of the cross-persistence diagrams
  9. Distinguishing Point Clouds via the Density of Cross-Persistence Diagrams (MTD)
  10. Experiments
  11. Distinction with simple test
  12. Applying noise for amplification of accuracy
  13. High-complexity regime (CIFAR100)
  14. Cross-RipsNet
  15. Two types of Cross-RipsNet
  16. ...and 17 more sections

Key Result

Theorem 1

Let $n, k \geq 1$. Assume that $M$ and $N$ are real analytic compact $d$-dimensional connected submanifolds, possibly with boundaries, and that $\mathbf{X} \in M^n$, $\mathbf{Y} \in N^k$ are random variables with densities with respect to the Hausdorff measures $\mathcal{H}_{dn}$ and $\mathcal{H}_{d

Figures (20)

  • Figure 1: Standard pipeline for Cross-barcode vectorization
  • Figure 2: In these pictures, the density of $MTD(Q_1, Q_1)$ is represented by dashed lines, and all other densities are represented by continuous lines. For each dataset, there is only one picture presented with one core cloud.
  • Figure 3: Densities of $MTD(Q_1, Q_1)$ (dashed) and $MTD(Q_1, Q_s)$ (solid). As can be seen from these images, the probability that our approach suggests that the value of $MTD(Q_1, Q_k)$ was sampled from the same distribution is greater than the classical significance threshold (0.05).
  • Figure 4: Densities of $MTD(Q_1, Q_1)$ (dashed) and $MTD(Q_1, Q_s)$ (solid), where Gaussian noise is applied only to the right argument (i.e., $MTD(\text{Pure}_i, \text{Noised}_i)$). From left to right, the relative noise norms $||\xi|| / ||x||$ are $[0\%, 25\%, 50\%, 75\%]$. As the noise intensity increases, the density of $MTD(Q_1, Q_1)$ gradually shifts to the right and eventually merges with the others once the objects become indistinguishable.
  • Figure 5: The process of adding Gaussian noise to COIL20 images. From left to right, the relative noise levels $||\xi|| / ||x||$ are $[0\%, 25\%, 50\%, 75\%]$
  • ...and 15 more figures

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Theorem 7: Coarea formula
  • Theorem 8
  • ...and 5 more