Related papers: Minimization of the Probabilistic p-frame Potentia…
Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…
A typical computational geometry problem begins: Consider a set P of n points in R^d. However, many applications today work with input that is not precisely known, for example when the data is sensed and has some known error model. What if…
The problem of searching for a model-based scene interpretation is analyzed within a probabilistic framework. Object models are formulated as generative models for range data of the scene. A new statistical criterion, the truncated object…
A new formalism is presented for analytically obtaining the probability density function, \( P_{n}(s) \), for the distance between two random points in an \( n \)-dimensional sphere of radius \( R \). Our formalism allows \( P_{n}(s) \) to…
We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere…
Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and…
How to distribute a set of points uniformly on a spherical surface is a very old problem that still lacks a definite answer. In this work, we introduce a physical measure of uniformity based on the distribution of distances between points,…
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner…
This paper studies probabilistic dual frames and the associated dual frame potentials from the perspective of optimal mass transport. The main contribution of this work shows that given a probabilistic frame, its associated dual frame…
We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for…
This paper concerns the geometric structure of optimizers for frame potentials. We consider finite, real or complex frames and rotation or unitarily invariant potentials, and mostly specialize to Parseval frames, meaning the frame potential…
Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using…
Herein, we address the expectations of frame potentials of three types of determinantal point processes(DPPs) on the d-dimensional unit sphere: (i) spherical ensembles on the 2-dimensional unit sphere; (ii) harmonic ensembles on the…
We have employed Particle Swarm Optimization to address a stochastic variant of the Smallest Enclosing Sphere estimation problem. An efficient algorithm has been developed to ascertain the optimal center and radius of a sphere encompassing…
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using…
This paper develops a systematic and geometric theory of optimal quantization on the unit sphere $\mathbb S^2$, focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional…
We propose a method to investigate modular structure in networks based on fitted probabilistic model, where the connection probability between nodes is related to a set of introduced local attributes. The attributes, as parameters of the…
Determination of \emph{optimal} arrangements of $N$ particles on a sphere is a well-known problem in physics. A famous example of such is the Thomson problem of finding equilibrium configurations of electrical charges on a sphere. More…