Related papers: Persistent homology and topological statistics of …
We study the probabilistic behavior of persistence-based statistics and propose a novel nonparametric framework for detecting structural changes in high-dimensional random point clouds. We establish moment bounds and tightness results for…
Topological defects are fundamental to the collective dynamics of non-equilibrium systems and in active matter, mediating spontaneous flows, dynamic self-organization, and emergent pattern formation. Here, we reveal critical states in…
Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description…
This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…
Hyperuniform systems are distinguished by an unusually strong suppression of large-scale density fluctuations and, consequently, display a high degree of uniformity at the largest length scales. In some cases, however, enhanced uniformity…
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
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…
Using a set of $\Lambda$CDM simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art tools of multiscale topological data analysis (TDA). We…
Disordered many-particle hyperuniform systems are exotic amorphous states characterized by anomalous suppression of large-scale density fluctuations. Here we substantially broaden the hyperuniformity concept along four different directions.…
Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…
Geometry and topology constitute complementary descriptors of three-dimensional shape, yet existing benchmark datasets primarily capture geometric information while neglecting topological structure. This work addresses this limitation by…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data,…
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…
While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations…
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…
By generalizing a class of models recently introduced to account for protracted transients in biological systems, we identify a novel mechanism for hyperuniformity. In this model, competition of particles over a shared resource guides the…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
We demonstrate that hyperuniformity, the suppression of density fluctuations at large length scales, emerges generically from the interplay between conservation laws and non-equilibrium driving. The underlying mechanism for this emergence…