Related papers: Discrete Morse Theory for Computing Zigzag Persist…
While normalizing flows for continuous data have been extensively researched, flows for discrete data have only recently been explored. These prior models, however, suffer from limitations that are distinct from those of continuous flows.…
This paper describes the adaptation of a well-scaling parallel algorithm for computing Morse-Smale segmentations based on path compression to a distributed computational setting. Additionally, we extend the algorithm to efficiently compute…
In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain…
In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded…
Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on…
We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X. Our approach is related to R. Forman's…
We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis,…
Extended and zigzag persistence were introduced more than ten years ago, as generalizations of ordinary persistence. While overcoming certain limitations of ordinary persistence, they both enjoy nice computational properties, which make…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain "Relative Morse Inequalities" relating the homology of the…
The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in TDA to summarise the shape of data. Crucial to their use is the bottleneck stability result. A generalisation of the Rips…
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and…
Persistent homology is a tool of topological data analysis that has been used in a variety of settings to characterize different dimensional holes in data. However, persistent homology computations can be memory intensive with a…
Mathematical morphology (MM) is a powerful and widely used framework in image processing. Through set-theoretic and discrete geometric principles, MM operations such as erosion, dilation, opening, and closing effectively manipulate digital…
In this paper, we extend the notion of directed clique complex to quivers and introduce an associated homology theory. By applying this construction to biquandle coloring quivers, we obtain new invariants of links. We then introduce a…