Related papers: Grounded persistent path homology: a stable, topol…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
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 topological (or graph) structures of real-world networks are known to be predictive of multiple dynamic properties of the networks. Conventionally, a graph structure is represented using an adjacency matrix or a set of hand-crafted…
The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e. a…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
Heterogeneous graphs offer powerful data representations for traffic, given their ability to model the complex interaction effects among a varying number of traffic participants and the underlying road infrastructure. With the recent advent…
Graph neural networks (GNNs) demonstrate a robust capability for representation learning on graphs with complex structures, showcasing superior performance in various applications. The majority of existing GNNs employ a graph convolution…
Topological data analysis (TDA) studies the shape patterns of data. Persistent homology is a widely used method in TDA that summarizes homological features of data at multiple scales and stores them in persistence diagrams (PDs). In this…
Geometric, topological and graph theory modeling and analysis of biomolecules are of essential importance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Magnitude homology is an emerging framework that captures the intrinsic topological and geometric features of metric spaces, demonstrating significant potential for topoplogical data analysis and geometric data analysis. This work…
Persistent Homology (PH) allows tracking homology features like loops, holes and their higher-dimensional analogs, along with a single-parameter family of nested spaces. Currently, computing descriptors for complex data characterized by…
A large driver of the complexity of graph learning is the interplay between structure and features. When analyzing the expressivity of graph neural networks, however, existing approaches ignore features in favor of structure, making it…
Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…
We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and…
In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic…
We present the analysis of the topological graph descriptor Local Degree Profile (LDP), which forms a widely used structural baseline for graph classification. Our study focuses on model evaluation in the context of the recently developed…
The machinery of topological data analysis becomes increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classification. Persistent homology is one of the key…
Graph Neural Networks (GNNs) have shown remarkable success across various scientific fields, yet their adoption in critical decision-making is often hindered by a lack of interpretability. Recently, intrinsically interpretable GNNs have…
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior…