An introduction to multiparameter persistence

MB Botnan, M Lesnick - arXiv preprint arXiv:2203.14289, 2022 - arxiv.org
In topological data analysis (TDA), one often studies the shape of data by constructing a
filtered topological space, whose structure is then examined using persistent homology …

Stable vectorization of multiparameter persistent homology using signed barcodes as measures

D Loiseaux, L Scoccola, M Carrière… - Advances in …, 2024 - proceedings.neurips.cc
Persistent homology (PH) provides topological descriptors for geometric data, such as
weighted graphs, which are interpretable, stable to perturbations, and invariant under, eg …

Generalized persistence diagrams for persistence modules over posets

W Kim, F Mémoli - Journal of Applied and Computational Topology, 2021 - Springer
When a category CC satisfies certain conditions, we define the notion of rank invariant for
arbitrary poset-indexed functors F: P → CF: P→ C from a category theory perspective. This …

[PDF][PDF] Signed barcodes for multi-parameter persistence via rank decompositions

MB Botnan, S Oppermann, S Oudot - … Geometry (SoCG 2022), 2022 - drops.dagstuhl.de
In this paper we introduce the signed barcode, a new visual representation of the global
structure of the rank invariant of a multi-parameter persistence module or, more generally, of …

Signed barcodes for multi-parameter persistence via rank decompositions and rank-exact resolutions

MB Botnan, S Oppermann, S Oudot - Foundations of Computational …, 2024 - Springer
In this paper, we introduce the signed barcode, a new visual representation of the global
structure of the rank invariant of a multi-parameter persistence module or, more generally, of …

Computing generalized rank invariant for 2-parameter persistence modules via zigzag persistence and its applications

TK Dey, W Kim, F Mémoli - Discrete & Computational Geometry, 2024 - Springer
The notion of generalized rank in the context of multiparameter persistence has become an
important ingredient for defining interesting homological structures such as generalized …

Homological approximations in persistence theory

B Blanchette, T Brüstle, EJ Hanson - Canadian Journal of …, 2024 - cambridge.org
We define a class of invariants, which we call homological invariants, for persistence
modules over a finite poset. Informally, a homological invariant is one that respects some …

[HTML][HTML] On the bottleneck stability of rank decompositions of multi-parameter persistence modules

MB Botnan, S Oppermann, S Oudot, L Scoccola - Advances in Mathematics, 2024 - Elsevier
A significant part of modern topological data analysis is concerned with the design and study
of algebraic invariants of poset representations—often referred to as persistence modules …

Approximation by interval-decomposables and interval resolutions of persistence modules

H Asashiba, EG Escolar, K Nakashima… - Journal of Pure and …, 2023 - Elsevier
In topological data analysis, two-parameter persistence can be studied using the
representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of …

Delaunay bifiltrations of functions on point clouds

ÁJ Alonso, M Kerber, T Lam, M Lesnick - … of the 2024 Annual ACM-SIAM …, 2024 - SIAM
Abstract The Delaunay filtration D.(X) of a point cloud X⊂ ℝd is a central tool of
computational topology. Its use is justified by the topological equivalence of D.(X) and the …