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Computation of $γ$-linear projected barcodes for multiparameter persistence

Alex Fernandes, Steve Oudot, Francois Petit

TL;DR

The paper addresses the challenge of representing multiparameter persistence with a discriminative yet computable descriptor by introducing the $\gamma$-linear projected barcode in the algebraic setting. It leverages conic-complexes and an augmented hyperplane arrangement to encode barcode templates across faces, enabling efficient queries along linear forms in the polar of a fixed cone $\gamma$. The approach unifies sheaf-theoretic and module-theoretic techniques to transfer persistence modules into a setting where derived pushforwards yield predictable barcodes, and extends the standard persistence algorithm to conic-complex filtrations. The results include a detailed algorithm, a complexity discussion in line with RIVET but with a simpler arrangement in practice, an illustrative example, and experiments validating performance on synthetic data. This work broadens the applicability of multiparameter persistence tools beyond functional settings and offers a practical, extensible framework for computing and querying $\gamma$-linear projected barcodes.

Abstract

The $γ$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone $γ$. So far, the computation of the $γ$-linear projected barcode has only been studied in the functional setting, in which persistence modules come from the persistent cohomology of $\mathbb{R}^n$-valued functions. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over $\mathbb{R}^n$ that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of $\mathbb{R}^n$ and a barcode template in each face of the arrangement; second, given any query linear form $u$ in the polar of $γ$, it locates $u$ within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.

Computation of $γ$-linear projected barcodes for multiparameter persistence

TL;DR

The paper addresses the challenge of representing multiparameter persistence with a discriminative yet computable descriptor by introducing the -linear projected barcode in the algebraic setting. It leverages conic-complexes and an augmented hyperplane arrangement to encode barcode templates across faces, enabling efficient queries along linear forms in the polar of a fixed cone . The approach unifies sheaf-theoretic and module-theoretic techniques to transfer persistence modules into a setting where derived pushforwards yield predictable barcodes, and extends the standard persistence algorithm to conic-complex filtrations. The results include a detailed algorithm, a complexity discussion in line with RIVET but with a simpler arrangement in practice, an illustrative example, and experiments validating performance on synthetic data. This work broadens the applicability of multiparameter persistence tools beyond functional settings and offers a practical, extensible framework for computing and querying -linear projected barcodes.

Abstract

The -linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone . So far, the computation of the -linear projected barcode has only been studied in the functional setting, in which persistence modules come from the persistent cohomology of -valued functions. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of and a barcode template in each face of the arrangement; second, given any query linear form in the polar of , it locates within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.
Paper Structure (32 sections, 23 theorems, 65 equations, 3 figures, 1 table)

This paper contains 32 sections, 23 theorems, 65 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Context
  3. Our contributions
  4. Preliminaries
  5. Free resolutions of persistence modules
  6. gamma-topology and gamma-sheaves
  7. From persistence modules to Alexandrov sheaves
  8. gamma-sheaves and Alexandrov sheaves
  9. One-dimensional persistence modules and gamma-linear projected barcode
  10. Conic complexes and their basic properties
  11. Definitions and basic properties
  12. Multiparameter conic complexes from free resolutions
  13. Direct images of finite conic-complexes
  14. From 1-d conic complexes to filtered cochain complexes
  15. From finite conic-complexes to filtered finite cochain complexes
  16. ...and 17 more sections

Key Result

Theorem 2.2

For any sheaf $\mathscr F$ in $\mathop{\mathrm{Mod}}\nolimits_{\mathbb{R}_c}(\mathbf{k}_\mathbb{R})$, there exists a unique locally finite multiset $\mathbf{B}(\mathscr F)$ of intervals of $\mathbb{R}$, called the barcode of $\mathscr F$, such that: $\mathscr F \simeq \bigoplus_{I \in \mathbf{B}(\ma

Figures (3)

  • Figure 1: The persistence module $M$
  • Figure 2: The point sample $P$.
  • Figure 3: The projected barcodes of the function-Rips bifiltration obtained by pushforwards along the linear forms $(x,y) \mapsto 0.875x+0.125y$ (left) and $(x,y) \mapsto 1x$ (right). Blue points represent persistence pairs with creator in degree 0 in the resolution of the first simplicial homology group, while red points represent persistence pairs with creator in degree 1 in the same resolution.

Theorems & Definitions (54)

  • Remark 2.1
  • Theorem 2.2: KS18
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 44 more