arXiv subjects

Michał Lipiński

Publications and source records attributed to Michał Lipiński.

13 recordsLinked to original sources

arXiv2026

Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions

Recent advances in combinatorial dynamical systems that generalize the classic discrete Morse theory have prompted algorithmic studies of combinatorial vector fields. In this regard, authors in [7] recently proposed the concept of Conley-Morse persistence barcode that summarizes the continuation of invariant sets in an evolving vector field through homological persistence. They proposed an algorithm to compute this barcode using a filtration of the so called \emph{index pairs} on a poset called \emph{transition diagram}. The algorithm becomes costly due to multiple runs of zigzag persistence it executes on filtrations of `unwieldy' structures of index pairs. We overcome this difficulty by replacing the index pairs with \emph{blocks}, which are structurally much simpler. These replacements need reversal of certain relations in the transition diagram resulting in a much simpler algorithm. The algorithm works by updating matrix decompositions akin to computing `vineyard' in standard persistence.

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arXiv2026

Morse theory for loop-free categories

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 introduce the notion of cellular categories. Second, we present a notion of vector field for loop-free categories. Third, we prove a homological collapsing theorem in the absence of critical objects in order to obtain the Morse inequalities. Examples are provided through the exposition. This answers partially a question by T. John: whether there is a Morse theory for loop-free (or acyclic) categories? [14].

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arXiv2026

The poset of cancellations induced by gradient dynamics in a filtered Lefschetz complex

Motivated by questions about simplification of topology, we take a discrete approach to the dependency of simplifying operations, using methods based on combinatorial gradient dynamics. We interpret the filter in persistent homology as a discrete Morse function. This lets us gradually simplify the dynamics in parallel with space and filter, while preserving homology. As a tool, we use shallow pairs, which are simultaneously birth-death pairs and combinatorial vectors. This allows us to extract topological features by the pairing of cells via persistence and simplify them using combinatorially defined cancellations. The main new concept is the depth poset of birth-death pairs, whose minimal elements are shallow pairs and whose linear extensions are sequences of cancellations that reduce the complex to its essential homology. Cancellations of birth-death pairs in a down set of this poset preserve the other birth-death pairs and the poset dependencies between them. An algorithm that constructs the depth poset in two passes of standard matrix reduction is given and proved correct.

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arXiv2026

Conley-Morse persistence barcode: a homological signature of combinatorial bifurcations

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 within the framework of combinatorial dynamical systems -- a young but already well-established theory. We introduce the Conley-Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of Conley-Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose.

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arXiv2026

Topological simplification guided by forbidden regions

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 cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c$\cdot$n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.

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arXiv2025

The Depth Poset under Transpositions in the Filter

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 the depth poset in the present work and for updating the persistence diagram under transpositions (Vineyard persistence), we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.

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arXiv2025

Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition

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 called a connection matrix.These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures - such as attractors, repellers, and orbits - in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in~\cite{DLMS24}. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of~\cite{DLMS24} and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.

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arXiv2024

Testing topological conjugacy of time series

This paper considers a problem of testing, from a finite sample, a topological conjugacy of two dynamical systems $(X,f)$ and $(Y,g)$. More precisely, given $x_1,\ldots, x_n \subset X$ and $y_1,\ldots,y_n \subset Y$ such that $x_{i+1} = f(x_i)$ and $y_{i+1} = g(y_i)$ as well as $h: X \rightarrow Y$, we deliver a number of tests to check if $f$ and $g$ are topologically conjugated via $h$. The values of the tests are close to zero for conjugated systems and large for systems that are not conjugated. Convergence of the test values, in case when sample size goes to infinity, is established. A number of numerical examples indicating scalability and robustness of the methods are given. In addition, we show how the presented method specialize to a test of sufficient embedding dimension in Takens' embedding theorem. Our methods also apply to the situation when we are given two observables of deterministic processes, of a form of one or higher dimensional time-series. In this case, their similarity can be accessed by comparing the dynamics of their Takens' reconstructions.

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arXiv2023

Morse predecomposition of an ivariant set

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 setting of combinatorial and classical dynamical systems. We prove that a Morse predecomposition indexed by a poset is a Morse decomposition and we show how a Morse predecomposition may be condensed back to a Morse decomposition.

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arXiv2023

Computing Connection Matrices via Persistence-like Reductions

Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Toward this goal, the classical theory for connection matrices has been adapted to combinatorial frameworks that facilitate computation. We develop an efficient persistence-like algorithm to compute a connection matrix from a given combinatorial (multi) vector field on a simplicial complex. This algorithm requires a single-pass, improving upon a known algorithm that runs an implicit recursion executing two-passes at each level. Overall, the new algorithm is more simple, direct, and efficient than the state-of-the-art. Because of the algorithm's similarity to the persistence algorithm, one may take advantage of various software optimizations from topological data analysis.

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arXiv2022

Tracking Dynamical Features via Continuation and Persistence

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 a method to track an isolated invariant set -- a salient feature of a combinatorial dynamical system -- across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.

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arXiv2020

Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces

We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in \cite{Mr2017}. The generalization consists in dropping the restrictive assumption in \cite{Mr2017} that every multivector has a unique maximal element. The extension is from the setting of Lefschetz complexes to the more general situation of finite topological spaces. We define isolated invariant sets, isolating neighbourhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities.

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arXiv2019

Persistence Bag-of-Words for Topological Data Analysis

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 structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.

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