Related papers: Curvature of high-dimensional data
Here we establish several results on the nonlocal curvature of planar curves. First we show how to express the nonlocal curvature of a curve relative to a point in terms of the nonlocal curvatures of simpler components of that curve…
A common observation in data-driven applications is that high-dimensional data have a low intrinsic dimension, at least locally. In this work, we consider the problem of point estimation for manifold-valued data. Namely, given a finite set…
In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our…
Point cloud surface reconstruction has improved in accuracy with advances in deep learning, enabling applications such as infrastructure inspection. Recent approaches that reconstruct from small local regions rather than entire point clouds…
In spite of considerable progress, computing curvature in Volume of Fluid (VOF) methods continues to be a challenge. The goal is to develop a function or a subroutine that returns the curvature in computational cells containing an interface…
We explore geometric aspects of bubble convergence for harmonic maps. More precisely, we show that the formation of bubbles is characterised by the local excess of curvature on the target manifold. We give a universal estimate for curvature…
Let $M$ denote a low-dimensional manifold embedded in Euclidean space and let ${X}= \{ x_1, \dots, x_n \}$ be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph…
In this paper, we propose using curvatures in digital space for 3D object analysis and recognition. Since direct adjacency has only six types of digital surface points in local configurations, it is easy to determine and classify the…
We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data…
The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be…
Density estimation is a crucial component of many machine learning methods, and manifold learning in particular, where geometry is to be constructed from data alone. A significant practical limitation of the current density estimation…
Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a…
The reconstruction of real-world surfaces is on high demand in various applications. Most existing reconstruction approaches apply 3D scanners for creating point clouds which are generally sparse and of low density. These points clouds will…
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes…
Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and…
Bump hunting deals with finding in sample spaces meaningful data subsets known as bumps. These have traditionally been conceived as modal or concave regions in the graph of the underlying density function. We define an abstract bump…
Model compression techniques allow to significantly reduce the computational cost associated with data processing by deep neural networks with only a minor decrease in average accuracy. Simultaneously, reducing the model size may have a…
Network data, commonly used throughout the physical, social, and biological sciences, consist of nodes (individuals) and the edges (interactions) between them. One way to represent network data's complex, high-dimensional structure is to…
This paper investigates a discretization scheme for mean curvature motion on point cloud varifolds with particular emphasis on singular evolutions. To define the varifold a local covariance analysis is applied to compute an approximate…
We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive…