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A Schauder Basis for Multiparameter Persistence

Peter Bubenik, Zachariah Ross

TL;DR

The paper addresses the challenge of representing multiparameter persistence by introducing signed barcodes on polyhedral pairs and embedding them into Banach spaces via a Schauder basis of compactly supported Lipschitz functionals built from nested CFK triangulations. The core method constructs a Schauder basis for $Lip_c(X,A)$ and uses it to define a vectorization map $F_{\mathbb{B}}$ into $\ell^1$ that is injective and Lipschitz-stable with respect to the (relative) 1-Wasserstein distance, $W_1$. The work generalizes from diagrams to relative Radon measures, proving stability and minimality results for the embedding, and provides both theoretical guarantees and intuitive visualizations of the vectorizations. The results enable statistically and machine-learning-friendly representations of multiparameter persistence, with precise bounds and a rigorous functional-analytic foundation for embeddings into sequence spaces and measure completions.

Abstract

Certain classes of multiparameter persistence modules may be encoded as signed barcodes, represented as points in a polyhedral subset of Euclidean space, we refer to as signed persistence diagrams. These signed persistence diagrams exist in the dual space of compactly supported, Lipschitz functionals on a polyhedral pair. In the interest of statistics and machine learning on multiparameter persistence modules, we aim to embed these signed persistence diagrams into Banach or Hilbert space. We use iteratively refined triangulations to define a Schauder Basis of compactly supported Lipschitz functionals. Evaluation of these functionals embeds signed persistence diagrams into the space of real-valued sequences. Furthermore, we show that in the larger space of relative Radon measures, the Schauder basis we have defined is minimal to induce an embedding.

A Schauder Basis for Multiparameter Persistence

TL;DR

The paper addresses the challenge of representing multiparameter persistence by introducing signed barcodes on polyhedral pairs and embedding them into Banach spaces via a Schauder basis of compactly supported Lipschitz functionals built from nested CFK triangulations. The core method constructs a Schauder basis for and uses it to define a vectorization map into that is injective and Lipschitz-stable with respect to the (relative) 1-Wasserstein distance, . The work generalizes from diagrams to relative Radon measures, proving stability and minimality results for the embedding, and provides both theoretical guarantees and intuitive visualizations of the vectorizations. The results enable statistically and machine-learning-friendly representations of multiparameter persistence, with precise bounds and a rigorous functional-analytic foundation for embeddings into sequence spaces and measure completions.

Abstract

Certain classes of multiparameter persistence modules may be encoded as signed barcodes, represented as points in a polyhedral subset of Euclidean space, we refer to as signed persistence diagrams. These signed persistence diagrams exist in the dual space of compactly supported, Lipschitz functionals on a polyhedral pair. In the interest of statistics and machine learning on multiparameter persistence modules, we aim to embed these signed persistence diagrams into Banach or Hilbert space. We use iteratively refined triangulations to define a Schauder Basis of compactly supported Lipschitz functionals. Evaluation of these functionals embeds signed persistence diagrams into the space of real-valued sequences. Furthermore, we show that in the larger space of relative Radon measures, the Schauder basis we have defined is minimal to induce an embedding.

Paper Structure

This paper contains 14 sections, 20 theorems, 58 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Signed Barcodes as Persistence Diagrams in Polyhedral Pairs
  4. Piece-wise Linear Triangulations
  5. Schauder Basis
  6. Relative Radon Measures
  7. Nested Triangulations
  8. Constructing a Schauder basis
  9. A Schauder Basis of Piece-wise Linear Functionals
  10. Stacked Functionals on Nested Coxeter-Freudenthal-Kuhn Triangulations
  11. Vectorizing Persistence Diagrams by Schauder Bases
  12. Feature Maps for Signed Persistence Diagrams
  13. Visualizing the Feature Map on Signed Persistence Diagrams
  14. Generalizing to Relative Radon Measures

Key Result

Lemma 2.10

PeterAlex(Lemma 6.1d) If $\pi$ is a coupling of $\alpha, \beta \in \hat{\mathcal{M}}_1^+(X,A)$, and $f,g \in Lip(X,A)$, then $\pi(f \oplus g) = \alpha(f) + \beta(g)$.

Figures (13)

  • Figure 1: Left: Point cloud $X \subset [0,1]^2$; Middle: Persistence diagram $\alpha$ in homology degree 1; Right: Visualization of $F_\mathbb{B}(\alpha)$
  • Figure 2: Left: Point cloud $X \subset [0,1]^2$; Middle:$2-$ parameter signed barcode for the density-rips filtration, generated by the Multipers package (Loiseaux and Schreiber, 2024) multipersRight: Visualization of $F_\mathbb{B}(\alpha)$ in $\mathbb{R}^3$
  • Figure 3: Left:Two classes of data, compared with a mixup barcode: Right Mixup Diagram of this pair of data. The dotted line represents the "mixup" of the feature
  • Figure 4: A triangulated approximation of a sphere, $X$, with closed subset $A$ consisting of a single point at the center
  • Figure 5: The Coxeter Freudenthal Kuhn triangulation at scale 1 on polyhedral pair $(\mathbb{R}^2_\leq, \Delta)$
  • ...and 8 more figures

Theorems & Definitions (50)

  • Example 2.1
  • Example 2.2
  • Example 2.3: Mixup Barcodes
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Lemma 2.10
  • ...and 40 more