Related papers: Stability of persistence spaces of vector-valued c…
In Topological Data Analysis, a common way of quantifying the shape of data is to use a persistence diagram (PD). PDs are multisets of points in $\mathbb{R}^2$ computed using tools of algebraic topology. However, this multi-set structure…
Algebraic persistence studies persistence modules (typically, linear representations of the poset $\mathbf{R}^n$ with $n \geq 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of…
We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension Z/2 by the fundamental group. By comparison with the space of…
In this paper, we study properties and patterns on permutations of multisets whose multivariate generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a property or pattern as…
A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global…
When persistence diagrams are formalized as the Mobius inversion of the birth-death function, they naturally generalize to the multi-parameter setting and enjoy many of the key properties, such as stability, that we expect in applications.…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data…
Given a pointwise finite-dimensional persistence module over a totally ordered set $S$, a theorem of Crawley-Boevey guarantees the existence of a barcode. When the set $S$ is finite, the persistence module is an equioriented type-A quiver…
We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems.…
The combinatorial interpretation of the persistence diagram as a M\"obius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a…
We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of…
Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher…
In this article, we derive a common fixed point result for a pair of single valued and set-valued mappings on a metric space having graphical structure. In this case, the set-valued map is assumed to be closed valued instead of closed and…
The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of…
Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self-homeomorphisms of a…
Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram…
Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields $\tau\in…
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…
We develop a new method to estimate the area, and more generally the intrinsic volumes, of a compact subset $X$ of $\mathbb{R}^d$ from a set $Y$ that is close in the Hausdorff distance. This estimator enjoys a linear rate of convergence as…