28 citations · 31 across the 8 of their papers we have counts for
15 papers
Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions
Tamal K. Dey, Andrew Haas, Michał Lipiński
Recent advances in combinatorial dynamical systems that generalize the classic discrete Morse theory have prompted algorithmic studies of combinatorial vector fields. In this regar…
Topological simplification guided by forbidden regions
Jakub Leśkiewicz, Bartosz Furmanek, Michał Lipiński +1
Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders c…
The Depth Poset under Transpositions in the Filter
Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek +2
The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing…
Conley-Morse persistence barcode: a homological signature of combinatorial bifurcations
Tamal K. Dey, Michał Lipiński, Manuel Soriano-Trigueros
Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations…
Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition
Tamal K. Dey, Michał Lipiński, Andrew Haas
Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is…
Morse predecomposition of an ivariant set
Michał Lipiński, Konstantin Mischaikow, Marian Mrozek
Motivated by the study of the recurrent orbits in a Morse set of a Morse decomposition, we introduce the concept of Morse predecomposition of an isolated invariant set in the setti…
The poset of cancellations induced by gradient dynamics in a filtered Lefschetz complex
Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek +1
Motivated by questions about simplification of topology, we take a discrete approach to the dependency of simplifying operations, using methods based on combinatorial gradient dyna…
Computing Connection Matrices via Persistence-like Reductions
Tamal K. Dey, Michał Lipiński, Marian Mrozek +1
Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for…
Testing topological conjugacy of time series
Paweł Dłotko, Michał Lipiński, Justyna Signerska-Rynkowska
This paper considers a problem of testing, from a finite sample, a topological conjugacy of two dynamical systems and . More precisely, given $x_1,\ldots, x_n \subse…
Tracking Dynamical Features via Continuation and Persistence
Tamal K. Dey, Michał Lipiński, Marian Mrozek +1
Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop…
Morse theory for loop-free categories
Michał Lipiński, David Mosquera-Lois, Mateusz Przybylski
We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen's Theorem A in this context and in…
Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces
Michał Lipiński, Jacek Kubica, Marian Mrozek +1
We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in \cite{Mr2017}. The generalization consists in dropping the restrictive as…
Persistence Bag-of-Words for Topological Data Analysis
Bartosz Zieliński, Michał Lipiński, Mateusz Juda +2
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex struct…
Persistence Codebooks for Topological Data Analysis
Bartosz Zielinski, Michal Lipinski, Mateusz Juda +2
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. The…
Persistent Homology of Morse Decompositions in Combinatorial Dynamics
Tamal K. Dey, Mateusz Juda, Tomasz Kapela +3
We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and…