Related papers: From Samples to Persistent Stratified Homotopy Typ…
Modern business and economic datasets often exhibit nonlinear, multi-scale structures that traditional linear tools under-represent. Topological Data Analysis (TDA) offers a geometric lens for uncovering robust patterns, such as connected…
Under the banner of `Big Data', the detection and classification of structure in extremely large, high dimensional, data sets, is, one of the central statistical challenges of our times. Among the most intriguing approaches to this…
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…
This paper lays the foundations of an approach to applying Gromov's ideas on quantitative topology to topological data analysis. We introduce the "contiguity complex", a simplicial complex of maps between simplicial complexes defined in…
It is shown that space-time may possess the differentiability properties of manifolds as well as the ultraviolet finiteness properties of lattices. Namely, if a field's amplitudes are given on any sufficiently dense set of discrete points…
We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney…
When analyzing parametric statistical models, a useful approach consists in modeling geometrically the parameter space. However, even for very simple and commonly used hierarchical models like statistical mixtures or stochastic deep neural…
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…
This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the…
This application paper presents a comprehensive experimental evaluation of the suitability of Topological Data Analysis (TDA) for the quantitative comparison of turbulent flows. Specifically, our study documents the usage of the persistence…
Science and engineering problems subject to uncertainty are frequently both computationally expensive and feature nonsmooth parameter dependence, making standard Monte Carlo too slow, and excluding efficient use of accelerated uncertainty…
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…
In this paper, we mainly build up the theory of sheaf-correspondence filtered spaces and stratified de Rham complexes for studying singular spaces. We prove the finiteness of a stratified de Rham cohomology and obtain its isomorphism to…
We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable $\infty$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem…
Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA --…
Locally convex (or nondegenerate) curves in the sphere (or projective space) have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames allows us to translate such curves into…
This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water…
In this paper, we introduce the persistence transformation, a novel methodology in Topological Data Analysis (TDA) for applications in time series data which can be obtained in various areas such as science, politics, economy, healthcare,…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
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…