Related papers: Statistical analysis of Mapper for stochastic and …
This paper employs techniques from metric geometry and optimal transport theory to address questions related to the analysis of functional data on metric or metric-measure spaces, which we refer to as fields. Formally, fields are viewed as…
Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar-valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to…
Determination of design parameters based on electromagnetic simulations of microwave circuits is an iterative and often time-consuming procedure. Space mapping is a powerful technique to optimize such complex models by efficiently…
Techniques from metric geometry have become fundamental tools in modern mathematical data science, providing principled methods for comparing datasets modeled as finite metric spaces. Two of the central tools in this area are the…
We study the question of approximating a compact geodesic metric space by metric graphs satisfying a uniform upper bound on their first Betti number. We prove that, up to a suitable multiplicative constant, Reeb graphs of distance functions…
Reeb spaces of continuous real-valued functions on topological spaces are fundamental and strong tools in investigating the spaces. The Reeb space is the natural quotient space of the space of the domain represented by connected components…
A labeled metric space is intuitively speaking a metric space together with a special set of points to be understood as the geometric boundary of the space. We study basic properties of a recently introduced labeled Gromov-Hausdorff…
We discuss scaling limits of large bipartite planar maps. If p is a fixed integer strictly greater than 1, we consider a random planar map M(n) which is uniformly distributed over the set of all 2p-angulations with n faces. Then, at least…
We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional…
Reeb graphs provide a method for studying the shape of a manifold by encoding the evolution and arrangement of level sets of a simple Morse function defined on the manifold. Since their introduction in computer graphics they have been…
Dimension reduction algorithms are a crucial part of many data science pipelines, including data exploration, feature creation and selection, and denoising. Despite their wide utilization, many non-linear dimension reduction algorithms are…
The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape…
An important problem in topological data analysis (TDA)$\unicode{x2014}$of both theoretical and practical interest$\unicode{x2014}$is to reconstruct the topology and geometry of an underlying (usually unknown) metric graph from possibly…
The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we introduce…
Given a continuous function $f:X\to\mathbb{R}$ and a cover $\mathcal{I}$ of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover $f^{-1}(\mathcal{I})$. Despite its success in applications, little is known…
The Reeb graph of a scalar function defined on a domain gives a topologically meaningful summary of that domain. Reeb graphs have been shown in the past decade to be of great importance in geometric processing, image processing, computer…
Performing inference in Bayesian models requires sampling algorithms to draw samples from the posterior. This becomes prohibitively expensive as the size of data sets increase. Constructing approximations to the posterior which are cheap to…
The Reeb graph is a construction which originated in Morse theory to study a real valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a…
Bayesian filtering is a key tool in many problems that involve the online processing of data, including data assimilation, optimal control, nonlinear tracking and others. Unfortunately, the implementation of filters for nonlinear, possibly…
Robust estimation of location is a fundamental problem in statistics, particularly in scenarios where data contamination by outliers or model misspecification is a concern. In univariate settings, methods such as the sample median and…