30,591 research outputs found
Orthogonal weighted linear L1 and L∞ approximation and applications
AbstractLet S={s1,s2,...,sn} be a set of sites in Ed, where every site si has a positive real weight ωi. This paper gives algorithms to find weighted orthogonal L∞ and L1 approximating hyperplanes for S. The algorithm for the weighted orthogonal L1 approximation is shown to require O(nd) worst-case time and O(n) space for d ≥ 2. The algorithm for the weighted orthogonal L∞ approximation is shown to require O(n log n) worst-case time and O(n) space for d = 2, and O(n⌊dl2 + 1⌋) worst-case time and O(n⌊(d+1)/2⌋) space for d > 2. In the latter case, the expected time complexity may be reduced to O(n⌊(d+1)/2⌋). The L∞ approximation algorithm can be modified to solve the problem of finding the width of a set of n points in Ed, and the problem of finding a stabbing hyperplane for a set of n hyperspheres in Ed with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the L∞ approximation algorithm
Numerical Homogenization of Heterogeneous Fractional Laplacians
In this paper, we develop a numerical multiscale method to solve the
fractional Laplacian with a heterogeneous diffusion coefficient. When the
coefficient is heterogeneous, this adds to the computational costs. Moreover,
the fractional Laplacian is a nonlocal operator in its standard form, however
the Caffarelli-Silvestre extension allows for a localization of the equations.
This adds a complexity of an extra spacial dimension and a singular/degenerate
coefficient depending on the fractional order. Using a sub-grid correction
method, we correct the basis functions in a natural weighted Sobolev space and
show that these corrections are able to be truncated to design a
computationally efficient scheme with optimal convergence rates. A key
ingredient of this method is the use of quasi-interpolation operators to
construct the fine scale spaces. Since the solution of the extended problem on
the critical boundary is of main interest, we construct a projective
quasi-interpolation that has both and dimensional averages over
subsets in the spirit of the Scott-Zhang operator. We show that this operator
satisfies local stability and local approximation properties in weighted
Sobolev spaces. We further show that we can obtain a greater rate of
convergence for sufficient smooth forces, and utilizing a global
projection on the critical boundary. We present some numerical examples,
utilizing our projective quasi-interpolation in dimension for analytic
and heterogeneous cases to demonstrate the rates and effectiveness of the
method
A comprehensive study of sparse codes on abnormality detection
Sparse representation has been applied successfully in abnormal event
detection, in which the baseline is to learn a dictionary accompanied by sparse
codes. While much emphasis is put on discriminative dictionary construction,
there are no comparative studies of sparse codes regarding abnormality
detection. We comprehensively study two types of sparse codes solutions -
greedy algorithms and convex L1-norm solutions - and their impact on
abnormality detection performance. We also propose our framework of combining
sparse codes with different detection methods. Our comparative experiments are
carried out from various angles to better understand the applicability of
sparse codes, including computation time, reconstruction error, sparsity,
detection accuracy, and their performance combining various detection methods.
Experiments show that combining OMP codes with maximum coordinate detection
could achieve state-of-the-art performance on the UCSD dataset [14].Comment: 7 page
Sard's approximation processes and oblique projections
Three problems arising in approximation theory are studied. These problems have already been studied by Arthur Sard. The main goal of this paper is to use geometrical compatibility theory to extend Sard's results and get characterizations of the sets of solutions.Fil: Corach, Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Giribet, Juan Ignacio. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Maestripieri, Alejandra Laura. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
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