Related papers: Combining persistent homology and invariance group…
A theoretical framework is presented for a (copula-based) notion of dissimilarity between continuous random vectors and its main properties are studied. The proposed dissimilarity assigns the smallest value to a pair of random vectors that…
Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity…
Persistent Homology (PH) is a fundamental tool in computational topology, designed to uncover the intrinsic geometric and topological features of data across multiple scales. Originating within the broader framework of Topological Data…
This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of…
Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such…
The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This…
We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980s with Vietoris-Rips Persistent Homology. For given integers $k\geq 0$ and $n\geq 1$ we consider the dimension $k$…
We introduce a general method for achieving robust group-invariance in group-equivariant convolutional neural networks ($G$-CNNs), which we call the $G$-triple-correlation ($G$-TC) layer. The approach leverages the theory of the…
We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group…
This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their…
In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud…
For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often…
Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative…
In recent years, networks with higher-order interactions have emerged as a powerful tool to model complex systems. Comparing these higher-order systems remains however a challenge. Traditional similarity measures designed for pairwise…
We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…
We present a robust method to find region-level correspondences between shapes, which are invariant to changes in geometry and applicable across multiple shape representations. We generate simplified shape graphs by jointly decomposing the…
Prediction and discovery of new materials with desired properties are at the forefront of quantum science and technology research. A major bottleneck in this field is the computational resources and time complexity related to finding new…
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a tuning parameter. Features with short…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…