Related papers: Decomposition of Persistent Homology and Spectral …
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such…
Understanding the magnetic fields of the Sun is essential for unraveling the underlying mechanisms driving solar activity. Integrating topological data analysis techniques into these investigations can provide valuable insights into the…
We investigate the similarities between adic finiteness and homological finiteness for chain complexes over a commutative noetherian ring. In particular, we extend the isomorphism properties of certain natural morphisms from homologically…
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data…
Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using…
Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
The aim of this paper is to introduce and study a geometric spectral sequence in Khovanov homology. The construction was motivated by a similar spectral sequence from Khovanov homology to Heegaard Floer homology.
Mapping between sequence and structure is currently an open problem in structural biology. Despite many experimental and computational efforts it is not clear yet how the structure is encoded in the sequence. Answering this question may…
An Important tool in the field topological data analysis is known as persistent Homology (PH) which is used to encode abstract representation of the homology of data at different resolutions in the form of persistence diagram (PD). In this…
Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic,…
Inspired by Rumin's work on a subcomplex in sub-Riemannian manifolds which is cohomologically equivalent to the de Rham complex, we present a more general construction that produces subcomplexes from any filtered cochain complex of finite…
We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived…
We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is…
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…
Zigzag persistent homology is a powerful generalisation of persistent homology that allows one not only to compute persistence diagrams with less noise and using less memory, but also to use persistence in new fields of application.…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
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…
We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into…
We study a variation of Carlsson and Zomorodian's persistent homology. As an application, we analyze the persistent cellular sheaf cohomology of network coding sheaves on deteriorating networks.