Abstract not found

Abstract not found

Abstract not found

Our objective is to study nonlinear approximation with regard to redundant systems. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. Greedy type approximations proved to be convenient and efficient ways of constructing m-term approximants. We introduce and study vector greedy algorithms that are designed with aim of constructing mth greedy approximants simultaneously for a given finite number of elements. We prove convergence theorems and obtain some estimates for the rate of convergence of vector greedy algorithms when elements come from certain classes.

Tree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual n-term approximation. We show that the restrictions of tree approximation cost little in terms of rates of approximation. We then use that result to design encoders for compression. These encoders are universal (they apply to general functions) and progressive (increasing accuracy is obtained by sending bit stream increments). We show optimality of the encoders in the sense that they provide upper estimates for the Kolmogorov entropy of Besov balls. AMS subject classication: 41A25, 41A46, 65F99, 65N12, 65N55. Key Words: compression, n-term approximation, encoding, Kolmogorov entropy . 1 Introduction Wavelets are utilized in many applications including image/signal processing and numerical methods for PDEs. Their usefulness stems in part from the fact that they provide ...

this paper is to discuss the existence and the nature of maximal spaces in the context of nonlinear methods based on thresholding (or shrinkage) procedures. Before going further, some remarks should be made:

Radial basis functions are tools for reconstruction of multivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial differential equations by collocation. The resulting very large linear N x N systems require efficient techniques for their solution, preferably of O(N) or O(N log N) computational complexity. This contribution describes some special lines of research towards this future goal. Theoretical results are accompanied by numerical examples, and various open problems are pointed out.

The solution of operator equations with radial basis functions by collocation in scattered points leads to large linear systems which often are non-sparse and ill-conditioned. But one can try to use only a subset of the data for the actual collocation, leaving the rest of the data points for error checking. This amounts to finding "sparse" approximate solutions of general linear systems arising from collocation. This contribution proposes an adaptive greedy method with proven (but slow) linear convergence to the full solution of the collocation equations. The collocation matrix need not be stored, and the progress of the method can be controlled by a variety of parameters. Some numerical examples are given.

In honor of Steve Smale’s 75-th birthday with the warmest regards of the authors Let ρ be an unknown Borel measure defined on the space Z: = X × Y with X ⊂ IR d and Y = [−M,M]. Given a set z of m samples zi = (xi,yi) drawn according to ρ, the problem of estimating a regression function fρ using these samples is considered. The main focus is to understand what is the rate of approximation, measured either in expectation or probability, that can be obtained under a given prior fρ ∈ Θ, i.e. under the assumption that fρ is in the set Θ, and what are possible algorithms for obtaining optimal or semi-optimal (up to logarithms) results. The optimal rate of decay in terms of m is established for many priors given either in terms of smoothness of fρ or its rate of approximation measured in one of several ways. This optimal rate is determined by two types of results. Upper bounds are established using various tools in approximation such as entropy, widths, and linear and nonlinear approximation. Lower bounds are proved using Kullback-Leibler information together with Fano inequalities and a certain type of entropy. A distinction is drawn between algorithms which employ knowledge of the prior in the construction of the estimator and those that do not. Algorithms of the second type which are universally optimal for a certain range of priors are given. 1

The question of finding an optimal dictionary for nonlinear m-term approximation is studied in the paper. We consider this problem in the periodic multivariate (d variables) case for classes of functions with mixed smoothness. We prove that the well known dictionary U d which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthonormal dictionaries. Next, it is established that for these classes near best m-term approximation with regard to U d can be achieved by simple greedy type (thresholding type) algorithm. The univariate dictionary U is used to construct a dictionary which is optimal among dictionaries with the tensor product structure.