Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Projected distances for multi-parameter persistence modules

Nicolas Berkouk, Francois Petit

TL;DR

The paper tackles the challenge of comparing multi-parameter persistence modules by bridging persistence with microlocal sheaf theory. It introduces projected barcodes, obtained by pushing a $\gamma$-sheaf forward to $\mathbb R$, and defines two families of distances, the integral sheaf metrics (ISM) and the sliced convolution distances (SCD), to quantify differences through one-parameter reductions. It proves stability results, relates projected barcodes to the classical fibered barcode, and shows that linear and $\gamma$-linear projections can be computed with standard one-parameter TDA software, offering efficiency advantages over full multi-parameter constructions. The framework also includes a rich theory for gamma-sheaves, sublevel-set persistence, and a family of projected-barcode metrics, providing practical tools for robust comparison and visualization of multi-parameter data. Overall, the approach delivers computable, stable invariants for multi-parameter persistence with clear connections to existing one-parameter TDA tooling and potential for scalable analysis.

Abstract

Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\mathbb{R}$. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the $γ$-linear ISM and the $γ$-linear SCD which are projected distances tailored for $γ$-sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous since our approach does not require computing nor storing an entire $n$-persistence module.

Projected distances for multi-parameter persistence modules

TL;DR

The paper tackles the challenge of comparing multi-parameter persistence modules by bridging persistence with microlocal sheaf theory. It introduces projected barcodes, obtained by pushing a -sheaf forward to , and defines two families of distances, the integral sheaf metrics (ISM) and the sliced convolution distances (SCD), to quantify differences through one-parameter reductions. It proves stability results, relates projected barcodes to the classical fibered barcode, and shows that linear and -linear projections can be computed with standard one-parameter TDA software, offering efficiency advantages over full multi-parameter constructions. The framework also includes a rich theory for gamma-sheaves, sublevel-set persistence, and a family of projected-barcode metrics, providing practical tools for robust comparison and visualization of multi-parameter data. Overall, the approach delivers computable, stable invariants for multi-parameter persistence with clear connections to existing one-parameter TDA tooling and potential for scalable analysis.

Abstract

Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto . Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the -linear ISM and the -linear SCD which are projected distances tailored for -sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous since our approach does not require computing nor storing an entire -persistence module.
Paper Structure (34 sections, 81 theorems, 204 equations, 6 figures, 1 table)

This paper contains 34 sections, 81 theorems, 204 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Sheaves
  3. Composition of kernels and integral transforms
  4. Convolution distance
  5. Constructible sheaves up to infinity
  6. gamma-sheaves
  7. Distances comparison
  8. The dimension one case
  9. Fibered Barcode
  10. Linear dimensionality reduction
  11. Linear direct image of sheaves
  12. Linear direct image of gamma-sheaves
  13. Sublevel sets persistence
  14. Properties of gamma-compactly generated sheaves
  15. Projected barcodes
  16. ...and 19 more sections

Key Result

Theorem 2.2

Let $X_i$ ($i=1,2,3,4$), be four $C^\infty$-manifolds and let $K_i\in\mathsf{D}^{\mathrm{b}}({\mathbf{k}}_{X_{i,i+1}})$, ($i=1,2,3$). Assume that $K_1$ is cohomologically constructible, $q_1$ is proper on $\mathop{\mathrm{Supp}}\nolimits(K_1)$ and $\mathrm{SS}(K_1)\cap (T^*_{X_1}X_1\times T^*X_2)\su

Figures (6)

  • Figure 1: Illustration of the push function $p_{\mathcal{L}_h}$
  • Figure 2: The sets $X$ (left) and $Y_s$ (right)
  • Figure 3: The sheaves $F$ and $G$ in $\mathsf{D}^{\mathrm{b}}_{{\mathbb{R}\rm{c}}}({\mathbf{k}}_{\mathbb{R}^2})$.
  • Figure 4: Illustration of the sets $R_I$
  • Figure 5: The datasets $X$ (left) and $Y$ (right)
  • ...and 1 more figures

Theorems & Definitions (167)

  • Example 2.1
  • Theorem 2.2: PS20
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8: KS18
  • ...and 157 more