| Constr. Approx. CONSTRUCTIVE APPROXIMATION c ○ 1994 Springer-Verlag NewYork Inc. Rates of Convex Approximation in Non-Hilbert Spaces (2009) | |||||||||||||
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| This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis ” subset S of a function space. Specifically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including in particular a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, for Lp spaces with 1 < p < ∞, the O(n −1/2) bounds of Barron and Jones are recovered when p = 2. One motivation for the questions studied here arises from the area of “artificial neural networks, ” where the problem can be stated in terms of the growth in the number of “neurons ” (the elements of S) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more “robust ” (resistant to exemplar noise) Lp, 1 ≤ p < 2 norms. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces. 1. | |||||||||||||
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