Related papers: Spherical Principal Curves
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…
Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional…
We introduce a technique to solve numerically the relativistic Euler's equations in scenarios with spherical symmetry using the standard Smoothed Particles Hydrodynamics method in cartesian coordinates. This implementation allow us to…
Covariance representations are developed for the uniform distributions on the Euclidean spheres in terms of spherical gradients and Hessians. They are applied to derive a number of Sobolev type inequalities and to recover and refine the…
Physical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences. These naturally occurring vector fields are often subject to physical constraints, such as being curl-free or…
We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous…
We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from…
The Euclidean paradigm that spheres optimize mean curvature variational problems breaks down in the sub-Riemannian Heisenberg group: neither the Pansu sphere nor the Kor\'anyi sphere is optimal for the variational problems associated with…
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the…
We introduce a new class of autoregressive models for spherical time series, where the dimension of the spheres on which the observations of the time series are situated may be finite-dimensional or infinite-dimensional as in the case of a…
In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…
Spherical symmetry is ubiquitous in nature. It's therefore unfortunate that spherical system simulations are so hard, and require complete spheres with millions of interacting particles. Here we introduce an approach to model spherical…
Spectral dimensionality reduction algorithms are widely used in numerous domains, including for recognition, segmentation, tracking and visualization. However, despite their popularity, these algorithms suffer from a major limitation known…
We will consider the resolution of the 3D non linear wave equation under the assumption of spherical symmetry on the Euclidian space. For this purpose, we will build a non trivial measure on distributions such that there exists a set of…
Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible…
The manifold of empirical mean values of statistical data ad infinitum has a geometric shape that depends on the probability measure that governs the generating model. Large deviation theory produces entropy functions that depend on both…
We study the statistical geometry of random chords on n-dimensional spheres by deriving explicit analytical expressions for the chord length distribution and its associated structural properties. A critical threshold emerges at dimension…
Data sets tend to live in low-dimensional non-linear subspaces. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The symmetric Riemannian geometry setting can be suitable for a variety of…
We establish the existence of hypersurfaces with constant mean curvature and a prescribed boundary in Euclidean space, represented as radial graphs over domains of the unit sphere. Under the assumptions that the mean curvature of the…
The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to…