Related papers: Parallel computation of interval bases for persist…
We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
We introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. Our algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. We show how to…
The persistence barcode (equivalently, the persistence diagram), which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-parameter persistent…
While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not…
We show that every infinite zigzag persistence module decomposes into a direct sum of interval persistence modules.
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal…
The goal of this note is to define biparametric persistence diagrams for smooth generic mappings $h=(f,g):M\to V\cong \mathbb{R}^2$ for smooth compact manifold $M$. Existing approaches to multivariate persistence are mostly centered on the…
Exterior powers play important roles in persistent homology in computational geometry. In the present paper we study the problem of extracting the $K$ longest intervals of the exterior-power layers of a tame persistence module. We prove a…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
Persistent homology (PH) is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general…
In this paper, we introduce a novel persistence framework for Morse decompositions in Markov chains using combinatorial multivector fields. Our approach provides a structured method to analyze recurrence and stability in finite-state…
Montgomery modular multiplication is widely-used in public key cryptosystems (PKC) and affects the efficiency of upper systems directly. However, modulus is getting larger due to the increasing demand of security, which results in a heavy…
0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of…
Finitely generated modules over the polynomial ring in $n$ indeterminates are isomorphic to quotients of finite rank free modules. We introduce a theory of relative Gr\"obner bases for those quotients of free modules and, equivalently, for…
Computations over the rational numbers often encounter the problem of intermediate coefficient growth. A solution to this is provided by modular methods, which apply the algorithm under consideration modulo a number of primes and then lift…
We define two notions. The first one is a $rank\ compression\ system$ $\xi$ for a finite poset $\mathbf{P}$ that assigns each interval subposet $I$ to an order-preserving map $\xi_I \colon I^{\xi} \to \mathbf{P}$ satisfying some conditions,…