Related papers: Persistent Cycle Representatives and Generalized L…
Recent years have witnessed an increased interest in the application of persistent homology, a topological tool for data analysis, to machine learning problems. Persistent homology is known for its ability to numerically characterize the…
While 2D occupancy maps commonly used in mobile robotics enable safe navigation in indoor environments, in order for robots to understand and interact with their environment and its inhabitants representing 3D geometry and semantic…
This paper introduces a reformulation of the classical convergence theorem for spectral sequences of filtered complexes which provides an algorithm to effectively compute the induced filtration on the total (co)homology, as soon as the…
It is crucial to learn the shared structures among functional predictors, as these structures characterize how predictor components exert common effects and, more generally, how predictors are homogeneously associated with the response.…
We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…
A universal cycle for a set S of combinatorial objects is a cyclic sequence of length |S|that contains a representation of each element in S exactly once as a substring. If S is the set of k-subsets of [n] = {1, 2, . . . , n}, it is…
Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to…
The effectiveness of Spatio-temporal Graph Neural Networks (STGNNs) in time-series applications is often limited by their dependence on fixed, hand-crafted input graph structures. Motivated by insights from the Topological Data Analysis…
Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first…
Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is…
In this paper we develop a novel Topological Data Analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a…
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…
A periodic cell complex, $K$, has a finite representation as the quotient space, $q(K)$, consisting of equivalence classes of cells identified under the translation group acting on $K$. We study how the Betti numbers and cycles of $K$ are…
We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
We consider the problem of representing multidimensional data where the domain of each dimension is organized hierarchically, and the queries require summary information at a different node in the hierarchy of each dimension. This is the…
We study rough high-dimensional landscapes in which an increasingly stronger preference for a given configuration emerges. Such energy landscapes arise in glass physics and inference. In particular we focus on random Gaussian functions, and…
Coupled natural systems are generally modeled at multiple abstraction levels. Both structural scale and behavioral complexity of these models are determinants in the kinds of questions that can be posed and answered. As scale and complexity…
The study of spanning trees and related structures is central in graph theory, closely connected to understanding functions between finite sets. This paper generalizes the established relationship between rooted trees and eventually…
We produce explicit geometric representatives of nontrivial homology classes in the space of long knots in R^d, when d is even. We generalize results of Cattaneo, Cotta-Ramusino and Longoni to define cycles which live off of the vanishing…