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Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of…

Graph pattern matching is often defined in terms of subgraph isomorphism, an NP-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be…

Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis -…

Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing…

We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…

Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…

Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and…

Topological features based on persistent homology capture high-order structural information so as to augment graph neural network methods. However, computing extended persistent homology summaries remains slow for large and dense graphs and…

Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than…

Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable,…

A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…

Long lived topological features are distinguished from short lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and…

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…

Persistent topological properties of an image serve as an additional descriptor providing an insight that might not be discovered by traditional neural networks. The existing research in this area focuses primarily on efficiently…

Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to…

In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…

Graph Laplacians as well as related spectral inequalities and (co-)homology provide a foray into discrete analogues of Riemannian manifolds, providing a rich interplay between combinatorics, geometry and theoretical physics. We apply some…

In this work we use the persistent homology method, a technique in topological data analysis (TDA), to extract essential topological features from the data space and combine them with deep learning features for classification tasks. In TDA,…

Scientific data has been growing in both size and complexity across the modern physical, engineering, life and social sciences. Spatial structure, for example, is a hallmark of many of the most important real-world complex systems, but its…