Related papers: Topo-Geometric Analysis of Variability in Point Cl…
We develop a language for describing the relationship among observations, mathematical models, and the underlying principles from which they are derived. Using Information Geometry, we consider geometric properties of statistical models for…
The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical MHD, it is important to verify that such simulations are in agreement with…
Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…
Persistent homology computes topological invariants from point cloud data. Recent work has focused on developing statistical methods for data analysis in this framework. We show that, in certain models, parametric inference can be performed…
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…
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…
Topological Data Analysis (TDA) has been applied with success to solve problems across many scientific disciplines. However, in the setting of a point cloud $X$ sampled from a shape $\mathcal{S}$ of low intrinsic dimension embedded within…
While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations…
Understanding the structure of high-dimensional data is fundamental to neuroscience and other data-intensive scientific fields. While persistent homology effectively identifies basic topological features such as "holes," it lacks the…
In this paper, we exploit minimal sensing information gathered from biologically inspired sensor networks to perform exploration and mapping in an unknown environment. A probabilistic motion model of mobile sensing nodes, inspired by motion…
Complex systems are difficult to study not only because they are nonlinear, multiscale, and often nonstationary, but because their scientifically relevant organization is often invisible at the level of individual components, pairwise…
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…
We present a topology-informed approach for classifying particle jets using persistent homology, a framework that captures the structural properties of point clouds. Particle jets produced in proton-proton collisions consist of cascades of…
Topological features play an essential role in ensuring geometric plausibility and structural consistency in image analysis tasks such as segmentation and skeletonization. However, integrating topology-preserving learning based on simple…
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…
Scientific data has been growing in both size and complexity across the modern physical, engineering, life and social sciences. Spatial structure, for example, is a hallmark of many of the most important real-world complex systems, but its…
Persistent homology has been widely used to study the topology of point clouds in $\mathbb{R}^n$. Standard approaches are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…
Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable,…
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and…