Nonlinear Lp approximation for 1 < p < 2

THE Lp - APPROXIMATION ORDER OF SURFACESPLINE INTERPOLATION FOR 1

Convex Polynomial and Spline Approximation in Lp , 0 < p <

We prove that a convex function f 6 L p [ -1 , 1], 0 < p < cx~, can be approximated by convex polynomials with an error not exceeding Cw~ (f, 1/n)p where w~(f, .) is the Ditzian-Totik modulus of smoothness of order three of f . We are thus filling the gap between previously known estimates involving og~(f, I/n)p, and the impossibility of having such estimates involving ~o4. We also give similar...

Restricted Nonlinear Approximation 1

The Robustness of Lp, p <1

In robust statistics, the breakdown point of an estimator is the percentage of outliers with which an estimator still generates reliable estimation. The upper bound of breakdown point is 50%, which means it is not possible to generate reliable estimation with more than half outliers [1-2] . In this paper, it is shown that for majority of experiences, when the outliers exceed 50%, but if they ar...

On Approximating the Lp Distances for p>2

Many applications in machine learning and data mining require computing pairwise lp distances in a data matrix A ∈ R . For massive high-dimensional data, computing all pairwise distances of A can be infeasible. In fact, even storing A or all pairwise distances of A in the memory may be also infeasible. For 0 < p ≤ 2, efficient small space algorithms exist, for example, based on the method of st...