Related papers: Efficient and Robust Persistent Homology for Measu…
Topological data analysis refers to approaches for systematically and reliably computing abstract ``shapes'' of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest…
High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the…
The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data…
This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each…
Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this…
Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we…
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially…
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
The \v{C}ech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech filtration, the number of simplices grows exponentially in the number of input points. A…
Magnitude homology is an emerging framework that captures the intrinsic topological and geometric features of metric spaces, demonstrating significant potential for topoplogical data analysis and geometric data analysis. This work…
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…
Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the…
Distance queries are a basic tool in data analysis. They are used for detection and localization of change for the purpose of anomaly detection, monitoring, or planning. Distance queries are particularly useful when data sets such as…
We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special…
We present Euler Characteristic Surfaces as a multiscale spatiotemporal topological summary of time series data encapsulating the topology of the system at different time instants and length scales. Euler Characteristic Surfaces with an…
Topological Data Analysis (TDA) is a novel statistical technique, particularly powerful for the analysis of large and high dimensional data sets. Much of TDA is based on the tool of persistent homology, represented visually via persistence…
Euclidean embeddings of data are fundamentally limited in their ability to capture latent semantic structures, which need not conform to Euclidean spatial assumptions. Here we consider an alternative, which embeds data as discrete…