Related papers: Persistent Cycle Representatives and Generalized L…

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of…

Persistent homology theory is a relatively new but powerful method in data analysis. Using simplicial complexes, classical persistent homology is able to reveal high dimensional geometric structures of datasets, and represent them as…

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 (PH) offers stable, multi-scale descriptors of intrinsic shape structure by capturing connected components, loops, and voids that persist across scales, providing invariants that complement purely geometric…

We give an $O(n^2(k+\log n))$ algorithm for computing the $k$-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on $n$ vertices. This is nearly quadratic in the number of vertices $n$, and therefore a…

The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex…

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point…

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…

The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of algebraic representation theory. In this work, we give a more accurate statement of the original theorem…

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,…

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit…

This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…

Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at…

An abstract simplicial complex is said to be $d$-representable if it records the intersection pattern of a collection of convex sets in $\mathbb{R}^d$. In this paper, we show that $d$-representability of a simplicial complex is equivalent…

We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally…

In this paper we follow up the study of 'complex complex landscapes,' rugged landscapes of many complex variables. Unlike real landscapes, the classification of saddles by index is trivial. Instead, the spectrum of fluctuations at…

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two…

Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…

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…

The perennial formalism is applied to the real, massive Klein-Gordon field on a globally-hyperbolic background space-time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two…