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Considerable eeort has been directed recently to develop asymptotically mini-max methods in problems of recovering innnite-dimensional objects (curves, densities , spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly-or exactly-minimax estimators being obtained for a variety of interesting problems.(More)
We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coeecients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel-and(More)
Density estimation is a commonly used test case for non-parametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coeecients. Minimax rates of convergence are studied over a large range of Besov function classes B s;p;q and for a range of global L 0 p error measures, 1 p 0 < 1. A single(More)
Deconvolution problems are naturally represented in the Fourier domain, whereas thresholding in wavelet bases is known to have broad adaptivity properties. We study a method which combines both fast Fourier and fast wavelet transforms and can recover a blurred function observed in white noise with O{n log.n/ 2 } steps. In the periodic setting, the method(More)
We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasi-exponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated(More)
Usually, methods for thresholding wavelet estimators are implemented term by term, with empirical coecients included or excluded depending on whether their absolute values exceed a level that re¯ects plausible moderate deviations of the noise. We argue that performance may be improved by pooling coecients into groups and thresholding them together. This(More)
Noise reduction by constrained reconstructions in the wavelet-transform domain. Department of Mathematics, Dart-NONLINEAR WAVELET METHODS 203 Andrew Bruce and Carl Taswell for many discussions about wavelet software. The NMR datasets were provided by Chris Raphael (Figure 1) and Jee Hoch (Figure 11), the ESCA dataset by Jean-Paul Bib erian, the image(More)
We consider the problem of estimating an unknown function f in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis {ψ jk (G), j, k} warped with the design. This allows to perform a very stable and computable thresholding algorithm. We investigate the properties of this new basis.(More)