41,462 research outputs found

    On Euclidean Norm Approximations

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    Euclidean norm calculations arise frequently in scientific and engineering applications. Several approximations for this norm with differing complexity and accuracy have been proposed in the literature. Earlier approaches were based on minimizing the maximum error. Recently, Seol and Cheun proposed an approximation based on minimizing the average error. In this paper, we first examine these approximations in detail, show that they fit into a single mathematical formulation, and compare their average and maximum errors. We then show that the maximum errors given by Seol and Cheun are significantly optimistic.Comment: 9 pages, 1 figure, Pattern Recognitio

    The geometry of optimal degree reduction of Bezier curves

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    Optimal degree reductions, i.e. best approximations of nn-th degree Bezier curves by Bezier curves of degree nn - 1, with respect to different norms are studied. It is shown that for any LpL_p-norm the euclidean degree reduction where the norm is applied to the euclidean distance function of two curves is identical to componentwise degree reduction. The Bezier points of the degree reductions are found to lie on parallel lines through the Bezier points of any Taylor expansion of degree nn - 1 of the original curve. This geometric situation is shown to hold also in the case of constrained degree reduction. The Bezier points of the degree reduction are explicitly given in the unconstrained case for pp = 1 and pp = 2 and in the constrained case for pp = 2

    Comments on "On Approximating Euclidean Metrics by Weighted t-Cost Distances in Arbitrary Dimension"

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    Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824-831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in Zn\mathbb{Z}^n. In this note we compare this approximation to two previously proposed Euclidean norm approximations and demonstrate that the empirical average errors given by Mukherjee are significantly optimistic in Rn\mathbb{R}^n. We also propose a simple normalization scheme that improves the accuracy of his approximation substantially with respect to both average and maximum relative errors.Comment: 7 pages, 1 figure, 3 tables. arXiv admin note: substantial text overlap with arXiv:1008.487

    Projecting Ising Model Parameters for Fast Mixing

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    Inference in general Ising models is difficult, due to high treewidth making tree-based algorithms intractable. Moreover, when interactions are strong, Gibbs sampling may take exponential time to converge to the stationary distribution. We present an algorithm to project Ising model parameters onto a parameter set that is guaranteed to be fast mixing, under several divergences. We find that Gibbs sampling using the projected parameters is more accurate than with the original parameters when interaction strengths are strong and when limited time is available for sampling.Comment: Advances in Neural Information Processing Systems 201
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