Corpus ID: 235829909

Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

  title={Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions},
  author={Magnus Bakke Botnan and Steffen Oppermann and Steve Oudot},
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 a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode encodes the rank invariant as a Z-linear combination of rank invariants of indicator modules supported on segments in the poset. It can also be enriched to encode the generalized rank invariant as a Z-linear… Expand
Amplitudes on abelian categories
The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that oneExpand
The Generalized Persistence Diagram Encodes the Bigraded Betti Numbers
  • Woojin Kim, Samantha Moore
  • Mathematics
  • 2021
We show that the generalized persistence diagram by Kim and Mémoli encodes the bigraded Betti numbers of finite 2-parameter persistence modules. More interestingly, the way to read off the bigradedExpand
$\ell^p$-Distances on Multiparameter Persistence Modules
It is shown that on 1or 2-parameter persistence modules over prime fields, dp I is the universal metric satisfying a natural stability property; this result extends a stability result of Skraba and Turner for the p-Wasserstein distance on barcodes in the 1- parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. Expand
On the consistency and asymptotic normality of multiparameter persistent Betti numbers
The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly byExpand


Generalized persistence diagrams for persistence modules over posets
The barcode of any interval decomposable persistence modules of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion, leading to a promotion of Patel's semicontinuity theorem about type $\mathcal{A}$ persistence diagram to Lipschitz continuity theorem for the category of sets. Expand
Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules
A generalization of the persistence algorithm based on a generalized matrix reduction technique that runs in $O(n^{2\omega})$ time where $\omega<2.373$ is the exponent for matrix multiplication. Expand
Interactive Visualization of 2-D Persistence Modules
The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way, with a novel data structure based on planar line arrangements. Expand
On Approximation of $2$D Persistence Modules by Interval-decomposables
In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. InExpand
The rank invariant stability via interleavings
A lower bound for the interleaving distance on persistence modules is given in terms of matching distance of rank invariants, and the internal stability of the rank invariant is proved in Terms of interleavings. Expand
Generalized persistence diagrams
  • Amit K. Patel
  • Mathematics, Computer Science
  • J. Appl. Comput. Topol.
  • 2018
The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer is generalized to the setting of constructible persistence modules valued in a symmetric monoidal category and a second type of persistence diagram is defined, which enjoys a stronger stability theorem. Expand
Multiparameter Persistence Image for Topological Machine Learning
This work introduces a new descriptor for multiparameter persistence, which it calls the Multiparameter Persistence Image, that is suitable for machine learning and statistical frameworks, is robust to perturbations in the data, has finer resolution than existing descriptors based on slicing, and can be efficiently computed on data sets of realistic size. Expand
Persistence Images: A Stable Vector Representation of Persistent Homology
This work converts a PD to a finite-dimensional vector representation which it is called a persistence image, and proves the stability of this transformation with respect to small perturbations in the inputs. Expand
Sliced Wasserstein Kernel for Persistence Diagrams
A new kernel for PDs is defined, which is not only provably stable but also discriminative (with a bound depending on the number of points in the PDs) w.r.t. the first diagram distance between PDs. Expand
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
  • M. Lesnick
  • Mathematics, Computer Science
  • Found. Comput. Math.
  • 2015
The theory of multidimensional interleavings is developed, with a view toward applications to topological data analysis, and it is shown that when the authors define their persistence modules over a prime field, d_\mathrm{I}$$dI satisfies a universality property. Expand