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Tracking the Persistence of Harmonic Chains: Barcode and Stability

Tao Hou, Salman Parsa, Bei Wang

TL;DR

The paper introduces the harmonic chain barcode, a stable, canonical topological descriptor that tracks the evolution of harmonic chains along filtrations, addressing non-uniqueness issues in ordinary persistence representations. It proves stability analogous to ordinary persistence and provides a cubic-time algorithm to compute the harmonic chain barcode on filtrations with $m$ simplices, enabling practical use in feature extraction and machine learning. By embedding the problem into a $\mathbb{U}$-indexed, block-decomposable framework and lifting to $\mathbb{R}^2$-indexed modules, it establishes a robust stability theory for sublevel-set filtrations and demonstrates a concrete interleaving-based bound on barcode perturbations. Overall, the harmonic chain barcode enriches persistent-homology descriptors with canonical representatives and stability guarantees, broadening potential applications in topology-driven data analysis and ML pipelines.

Abstract

The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of data, the persistence barcode tracks the evolution of its homology groups. In this paper, we introduce a new type of barcode, called the harmonic chain barcode, which tracks the evolution of harmonic chains. In addition, we show that the harmonic chain barcode is stable. Given a filtration of a simplicial complex of size $m$, we present an algorithm to compute its harmonic chain barcode in $O(m^3)$ time. Consequently, the harmonic chain barcode can enrich the family of topological descriptors in applications where a persistence barcode is applicable, such as feature vectorization and machine learning.

Tracking the Persistence of Harmonic Chains: Barcode and Stability

TL;DR

The paper introduces the harmonic chain barcode, a stable, canonical topological descriptor that tracks the evolution of harmonic chains along filtrations, addressing non-uniqueness issues in ordinary persistence representations. It proves stability analogous to ordinary persistence and provides a cubic-time algorithm to compute the harmonic chain barcode on filtrations with simplices, enabling practical use in feature extraction and machine learning. By embedding the problem into a -indexed, block-decomposable framework and lifting to -indexed modules, it establishes a robust stability theory for sublevel-set filtrations and demonstrates a concrete interleaving-based bound on barcode perturbations. Overall, the harmonic chain barcode enriches persistent-homology descriptors with canonical representatives and stability guarantees, broadening potential applications in topology-driven data analysis and ML pipelines.

Abstract

The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of data, the persistence barcode tracks the evolution of its homology groups. In this paper, we introduce a new type of barcode, called the harmonic chain barcode, which tracks the evolution of harmonic chains. In addition, we show that the harmonic chain barcode is stable. Given a filtration of a simplicial complex of size , we present an algorithm to compute its harmonic chain barcode in time. Consequently, the harmonic chain barcode can enrich the family of topological descriptors in applications where a persistence barcode is applicable, such as feature vectorization and machine learning.

Paper Structure

This paper contains 19 sections, 27 theorems, 57 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Harmonic Cycles
  5. Ordinary Persistence
  6. Zigzag Persistence
  7. Harmonic Chain Barcodes and Representatives
  8. A Cubic-Time Algorithm for Computing Harmonic Chain Barcodes
  9. Implementation
  10. Sublevel Set Harmonic Chain Barcode and Its Stability
  11. Interleaving and Stability for Ordinary Persistence
  12. Sub-level-Set Filtration
  13. $\mathbb{U}$-indexed Persistence Modules
  14. Blocks and Block-Decomposable Modules
  15. Interleaving of $\mathbb{U}$-Indexed Modules
  16. ...and 4 more sections

Key Result

Lemma 2

$\mathbb{H}{\hbox{$\mathbb{H}$}}_p(K)$ is isomorphic to $H_p(K)$ and $H^p(K)$. Specifically, each homology and cohomology class has a unique harmonic cycle in it.

Figures (4)

  • Figure 1: An example of the computation of harmonic chain barcode $\mathbb{H}{\hbox{$\mathbb{H}$}}_1(F)$ and representatives.
  • Figure 2: Harmonic chain barcode and ordinary persistence barcode for the filtration in \ref{['fig:filtbarc']}. Deviating from conventions in ordinary persistence, bars are drawn as closed integer intervals, e.g., $[7,7]$ in the ordinary barcode is killed by the addition of $abe$ in $K_8$.
  • Figure 3: Turning closed-open spans into closed bars
  • Figure 4: The extension of a closed interval is a closed block.

Theorems & Definitions (54)

  • Definition 1
  • Lemma 2: Eckmann1944
  • Proposition 3: DeSilvaMorozovVejdemoJohansson2011
  • Theorem 4
  • Proposition 5
  • proof
  • Definition 6
  • Theorem 7
  • proof
  • Definition 8: Harmonic chain barcode
  • ...and 44 more