Related papers: Topological comparison of some dimension reduction…

Persistence diagrams (PDs) are used as signatures of point cloud data. Two clouds of points can be compared using the bottleneck distance d_B between their PDs. A potential drawback of this pipeline is that point clouds sampled from…

Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their…

Spatial transcriptomics studies are becoming increasingly large and commonplace, necessitating simultaneous analysis of a large number of spatially resolved variables. Correspondingly, a diverse range of methodologies have been proposed to…

In the machine learning field, dimensionality reduction is an important task. It mitigates the undesired properties of high-dimensional spaces to facilitate classification, compression, and visualization of high-dimensional data. During the…

Topological data analysis (TDA) is a rising branch in modern applied mathematics. It extracts topological structures as features of a given space and uses these features to analyze digital data. Persistent homology, one of the central tools…

Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information,…

Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…

In this work, we explore dimensionality reduction techniques for univariate and multivariate time series data. We especially conduct a comparison between wavelet decomposition and convolutional variational autoencoders for dimension…

Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. The result is a space which requires less physical memory and has a faster distance calculation. These techniques are widely used…

Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics…

Finding meaningful distances between high-dimensional data samples is an important scientific task. To this end, we propose a new tree-Wasserstein distance (TWD) for high-dimensional data with two key aspects. First, our TWD is specifically…

Persistent homology has been widely used to discover hidden topological structures in data across various applications, including music data. To apply persistent homology, a distance or metric must be defined between points in a point cloud…

Nonlinear dimensionality reduction methods are a popular tool for data scientists and researchers to visualize complex, high dimensional data. However, while these methods continue to improve and grow in number, it is often difficult to…

This work introduces a number of algebraic topology approaches, such as multicomponent persistent homology, multi-level persistent homology and electrostatic persistence for the representation, characterization, and description of small…

This article studies the robust version of persistent homology based on trimming methodology to capture the geometric feature through support of the data in presence of outliers. Precisely speaking, the proposed methodology works when the…

We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to…

Multi-connected universe models with space identification scales smaller than the size of the observable universe produce topological images of cosmic sources. We generalise to locally hyperbolic spaces the crystallographic method, aimed 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…

In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes,…

Topological methods are very rarely used in structural health monitoring (SHM), or indeed in structural dynamics generally, especially when considering the structure and topology of observed data. Topological methods can provide a way of…