Projected distances for multi-parameter persistence modules
[Distances projetées pour les modules de persistance à plusieurs paramètres]
Berkouk, Nicolas1  ; Petit, François2
1Laboratory for Topology and Neuroscience, EPFL, Lausanne (Switzerland)
2Université Paris Cité and Université Sorbonne Paris Nord, Inserm, INRAE, Center for Research in Epidemiology and StatisticS (CRESS), 75004 Paris (France)
Annales de l'Institut Fourier, Online first, 62 p.

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 $\gamma $-linear ISM and the $\gamma $-linear SCD which are projected distances tailored for $\gamma $-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.

En nous appuyant sur la théorie des faisceaux, nous introduisons les notions de code-barres projetés et de distances projetées pour les modules de persistance à plusieurs paramètres. Les code-barres projetés sont définis comme le poussé en avant dérivé des modules de persistance sur $\mathbb{R}$. Les distances projetées viennent en deux familles : les métriques intégrales de faisceaux (ISM) et les distances de convolution tranchées (SCD). Nous menons une étude systématique de la stabilité des code-barres projetés et montrons que le code-barre fibré est une instance particulière des code-barres projetés. Nous prouvons que l’ISM et la SCD fournissent des bornes inférieures pour la distance de convolution. De plus, nous montrons que les ISM $\gamma $-linéaires et les SCD $\gamma $-linéaires, qui sont des distances projetées adaptées aux $\gamma $-faisceaux, peuvent être calculées à l’aide de logiciels de TDA dédiés aux modules de persistance à un paramètre. De plus, la complexité en temps et en mémoire requise pour calculer ces deux métriques est avantageuse, car notre approche ne nécessite ni le calcul ni le stockage d’un module de persistance à $n$ paramètres.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3752
Classification : 55N31, 55N30, 35A27
Keywords: topological data analysis, multi-parameter persistence, sheaf theory
Mots-clés : analyse de données topologique, persistance à plusieurs paramètres, théorie des faisceaux

Berkouk, Nicolas  1   ; Petit, François  2

1 Laboratory for Topology and Neuroscience, EPFL, Lausanne (Switzerland)
2 Université Paris Cité and Université Sorbonne Paris Nord, Inserm, INRAE, Center for Research in Epidemiology and StatisticS (CRESS), 75004 Paris (France) 
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Berkouk, Nicolas; Petit, François. Projected distances for multi-parameter persistence modules. Annales de l'Institut Fourier, Online first, 62 p.

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