Related papers: Defining and computing persistent Z-homology in th…
In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact…
The Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which…
In this work, we construct a persistent version of the well-known Leray spectral sequence. More precisely, we construct a spectral sequence that computes the persistent cohomology of a space from the persistent cohomology in each open set…
We study the persistent homology of the data set of syntactic parameters of the world languages. We show that, while homology generators behave erratically over the whole data set, non-trivial persistent homology appears when one restricts…
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…
The persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors…
One of the primary areas of interest in applied algebraic topology is persistent homology, and, more specifically, the persistence diagram. Persistence diagrams have also become objects of interest in topological data analysis. However,…
Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…
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…
Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often…
Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are…
Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evolution of the subject. The goal is to highlight the main ideas,…
Within the context of topological data analysis, the problems of identifying topological significance and matching signals across datasets are important and useful inferential tasks in many applications. The limitation of existing solutions…
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.…
A stable homology theory is defined for completely distributive CSL algebras in terms of the point-neighbourhood homology of the partially ordered set of meet-irreducible elements of the invariant projection lattice. This specialises to the…
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied…
Given a CW-complex A we define an `A-shaped' homology theory which behaves nicely towards A-homotopy groups allowing the generalization of many classical results. We also develop a relative version of the Federer spectral sequence for…
The literature in persistent homology often refers to a "structure theorem for finitely generated graded modules over a graded principal ideal domain". We clarify the nature of this structure theorem in this context.
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…