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The Delaunay-Rips filtration is a lighter and faster alternative to the well-known Rips filtration for low-dimensional Euclidean point clouds. Despite these advantages, it has seldom been studied. In this paper, we aim to bridge this gap by…
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…
A popular method for selecting the number of clusters is based on stability arguments: one chooses the number of clusters such that the corresponding clustering results are "most stable". In recent years, a series of papers has analyzed the…
Stability selection (Meinshausen and Buhlmann, 2010) makes any feature selection method more stable by returning only those features that are consistently selected across many subsamples. We prove (in what is, to our knowledge, the first…
Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations…
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…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
Latent space matching, which consists of matching distributions of features in latent space, is a crucial component for tasks such as adversarial attacks and defenses, domain adaptation, and generative modelling. Metrics for probability…
Many clustering algorithms when the data are curves or functions have been recently proposed. However, the presence of contamination in the sample of curves can influence the performance of most of them. In this work we propose a robust,…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
A new interpoint distance-based measure is proposed to identify the optimal number of clusters present in a data set. Designed in nonparametric approach, it is independent of the distribution of given data. Interpoint distances between the…
Model selection is a major challenge in non-parametric clustering. There is no universally admitted way to evaluate clustering results for the obvious reason that no ground truth is available. The difficulty to find a universal evaluation…
Clustering is an important tool for data exploration where the goal is to subdivide a data set into disjoint clusters that fit well into the underlying data structure. When dealing with sensitive data, privacy-preserving algorithms aim to…
Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding…
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to…
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…
In high-dimension, low-sample size (HDLSS) data, it is not always true that closeness of two objects reflects a hidden cluster structure. We point out the important fact that it is not the closeness, but the "values" of distance that…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space…
In urgent decision making applications, ensemble simulations are an important way to determine different outcome scenarios based on currently available data. In this paper, we will analyze the output of ensemble simulations by considering…