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Journals and Conferences
In this paper, we study the numerical integration of continuous functions on d-dimensional spheres S ⊆ R by equally weighted quadrature rules based at N ≥ 1 points on S which minimize a generalized energy functional. Examples of such points are configurations, which minimize energies for the Riesz kernel ‖x− y‖−s 0 < s ≤ d and logarithmic kernel − log ‖x−… (More)
Determining the intrinsic dimension of a hyperspectral image is an important step in the spectral unmixing process and under- or overestimation of this number may lead to incorrect unmixing in unsupervised methods. In this paper, we discuss a new method for determining the intrinsic dimension using recent advances in random matrix theory. This method is… (More)
In this short article, we explore some methods, results and open problems dealing with weighted Marcinkiewicz-Zygmund inequalities as well as the numerical approximation of integrals for exponential weights on the real line and on finite intervals of the line. The problems posed are based primarily on ongoing work of the author and his collaborators and… (More)
It is very difficult to analyze large amounts of hyperspectral data. Here we present a method based on reducing the dimensionality of the data and clustering the result in moving toward classification of the data. Dimensionality reduction is done with diffusion maps, which interpret the eigenfunctions of Markov matrices as a system of coordinates on the… (More)
We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1 ≤ k ≤ n. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes. 1991 AMS(MOS) Classification: Primary: 11T30; secondary: 05B15.
Criteria for optimally discretizing measurable sets in Euclidean space is a difficult and old problem which relates directly to the problem of good numerical integration rules or finding points of low discrepancy. On the other hand, learning meaningful descriptions of a finite number of given points in a measure space is an exploding area of research with… (More)
We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class A of d-dimensional compact sets embedded in R ′ , 1 ≤ d ≤ d′. We assume that these points interact through a Riesz potential V = | · |−s, where s > 0 and | · | is the Euclidean distance in R ′ . With δ∗ s (A,N) and ρ ∗ s(A,N)… (More)
+1 variables over finite fields to scatter points on the surface of the unit sphere S d , d≥1. Applications are given for spherical t designs and generalized s energies.
The presence of large amounts of data in hyperspectral images makes it very difficult to perform further tractable analyses. Here, we present a method of analyzing real hyperspectral data by dimensionality reduction using diffusion maps. Diffusion maps interpret the eigenfunctions of Markov matrices as a system of coordinates on the original data set in… (More)
Let (M, g1) be a compact d-dimensional Riemannian manifold. Let Xn be a set of n sample points in M drawn randomly from a Lebesgue density f . Define two points x, y ∈ M . We prove that the normalized length of the power-weighted shortest path between x, y passing through Xn converges to the Riemannian distance between x, y under the metric gp = f g1, where… (More)