Related papers: Principal Component Analysis of Persistent Homolog…
We study the long-standing problem of determining the number of principal components in econometric applications from a selective inference perspective. We consider i.i.d. observations from a $p$-dimensional random vector with $p<n$ and…
Proposing scoring functions to effectively understand, analyze and learn various properties of high dimensional hidden representations of large-scale transformer models like BERT can be a challenging task. In this work, we explore a new…
We present a machine learning approach that leverages persistent homology to classify bacterial flagellar motors into two functional states: rotated and stalled. By embedding protein structural data into a topological framework, we extract…
Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations)…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a…
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…
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…
Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend…
One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…
We propose a stable version of Principal Component Analysis (PCA) in the general framework of a separable Hilbert space. It consists in interpreting the projection on the first eigenvectors as a step function applied to the spectrum of the…
For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often…
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a tuning parameter. Features with short…
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…
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…
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such…
We consider change-point tests based on rank statistics to test for structural changes in long-range dependent observations. Under the hypothesis of stationary time series and under the assumption of a change with decreasing change-point…
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…
In order theory, a rank function measures the vertical "level" of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure…
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a…