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PH-STAT

Moo K. Chung

TL;DR

PH-STAT provides a MATLAB-based toolkit for statistical inference on persistent homology in brain networks, integrating Rips and graph filtrations, homology computations (including boundary operators and Hodge Laplacians), and scalable topological distances. Its key innovations include birth–death decomposition, topological embedding and averaging, a transposition-accelerated permutation test, and Wasserstein-based clustering, all designed for scalable analysis of large connectivity data. The toolbox emphasizes interpretability and practicality through visualization, documentation, and open-source availability, enabling researchers to derive topology-informed insights from diverse data types. Overall, PH-STAT offers a coherent, scalable framework that translates topological summaries into actionable brain-network diagnostics and comparisons across groups or conditions.

Abstract

We introduce PH-STAT, a comprehensive MATLAB toolbox designed for performing a wide range of statistical inferences and machine learning tasks on persistent homology, primarily for network and graph data, with an emphasis on brain network analysis. Persistent homology is a prominent tool in topological data analysis (TDA) that captures the underlying topological features of complex data sets. The toolbox aims to provide users with an accessible and user-friendly interface for analyzing and interpreting topological data. The Matlab package is distributed in https://github.com/laplcebeltrami/PH-STAT.

PH-STAT

TL;DR

PH-STAT provides a MATLAB-based toolkit for statistical inference on persistent homology in brain networks, integrating Rips and graph filtrations, homology computations (including boundary operators and Hodge Laplacians), and scalable topological distances. Its key innovations include birth–death decomposition, topological embedding and averaging, a transposition-accelerated permutation test, and Wasserstein-based clustering, all designed for scalable analysis of large connectivity data. The toolbox emphasizes interpretability and practicality through visualization, documentation, and open-source availability, enabling researchers to derive topology-informed insights from diverse data types. Overall, PH-STAT offers a coherent, scalable framework that translates topological summaries into actionable brain-network diagnostics and comparisons across groups or conditions.

Abstract

We introduce PH-STAT, a comprehensive MATLAB toolbox designed for performing a wide range of statistical inferences and machine learning tasks on persistent homology, primarily for network and graph data, with an emphasis on brain network analysis. Persistent homology is a prominent tool in topological data analysis (TDA) that captures the underlying topological features of complex data sets. The toolbox aims to provide users with an accessible and user-friendly interface for analyzing and interpreting topological data. The Matlab package is distributed in https://github.com/laplcebeltrami/PH-STAT.
Paper Structure (22 sections, 4 theorems, 80 equations, 24 figures)

This paper contains 22 sections, 4 theorems, 80 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Morse Filtration
  3. Simplical Homology
  4. Rips complex
  5. Boundary matrix
  6. Homology group
  7. Hodge Laplacian
  8. Rips filtrations
  9. Persistent diagrams
  10. Graph Filtration
  11. Graph filtration
  12. Monotone Betti curves
  13. Rips filtration vs. graph filtration
  14. Graph filtration in trees
  15. Birth-death decomposition
  16. ...and 7 more sections

Key Result

Theorem 4.1

The edge weight set $W = \{ w_{(1)}, \cdots, w_{(q)} \}$ has the unique decomposition where birth set $W_b = \{ b_{(1)}, b_{(2)}, \cdots, b_{(q_0)} \}$ is the collection of 0D sorted birth values and death set $W_d = \{ d_{(1)}, d_{(2)}, \cdots, d_{(q_1)} \}$ is the collection of 1D sorted death values with $q_0 = p-1$ and $q_1 = (p-1)(p-2)/2$. Further $W_b$ forms the 0D persistent d

Figures (24)

  • Figure 1: The births and deaths of connected components in the sublevel sets in a Morse filtration chung.2009.IPMI. We have local minimums $a < b < d <f$ and local maximums $c< e$. At $y=a$, we have a single connected component (gray area). As we increase the filtration value to $y=b$, we have the birth of a new component (second gray area). At the local maximum $y=c$, the two sublevel sets merge together to form a single component. This is viewed as the death of a component. The process continues till we exhaust all the critical values. Following the Elder rule, we pair birth to death: $(b,c)$ and $(d,e)$. Other critical values are paired similarly. These paired points form the persistent diagram.
  • Figure 2: Left: 50 randomly distributed points $X$ in $[0,1]^3$. Right: Rips complex $R_{0.3}(X)$ within radius 0.3 containing 106 1-simplices, 75 2-simplices (yellow) and 22 3-simplices (blue).
  • Figure 3: A simplicial complex with 5 vertices and 2-simplex $\sigma =[v_1, v_2, v_3]$ with a filled-in face (colored gray). After boundary operation $\partial_2$, we are only left with 1-simplices $[v_1, v_2] + [v_2, v_3] + [v_3, v_1]$, which is the boundary of the filled in triangle. The complex has a single connected component ($\beta_0=1$) and a single 1-cycle. The dotted red arrows are the orientation of simplices.
  • Figure 4: Examples of boundary matrix computation. From the left to right, the radius is changed to 0.5, 0.6 and 1.0.
  • Figure 5: The rank-nullity theorem for boundary matrix $\boldsymbol{\partial}_k$, which states the dimension of the domain of $\boldsymbol{\partial}_k$ is the sum of the dimension of its image and the dimension of its kernel (nullity).
  • ...and 19 more figures

Theorems & Definitions (9)

  • Example 2.1
  • Example 3.1
  • Example 3.2
  • Theorem 4.1: Birth-death decomposition
  • Theorem 6.1
  • Theorem 6.2
  • Theorem 8.1
  • proof
  • Example 8.1