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Minimax Estimation via Wavelet Shrinkage
, 1992
"... 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 coe cients. The shrinkage can be tuned to be nearly minim ..."
Abstract

Cited by 246 (32 self)
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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 coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel and Besovtype smoothness constraints, and asymptotically minimax over Besov bodies with p q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p <2, so our method can signi cantly outperform every linear method (kernel, smoothing spline, sieve,:::) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which mayvary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper discuss practical implementation, spatial adaptation properties and applications to inverse problems.
Unconditional bases are optimal bases for data compression and for statistical estimation
 Applied and Computational Harmonic Analysis
, 1993
"... An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and ..."
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Cited by 141 (23 self)
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An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and recovering than they do in any other orthogonal basis. In fact, simple thresholding in an unconditional basis works essentially better for recovery and estimation than other methods, period. (Performance is measured in an asymptotic minimax sense.) As an application, we formalize and prove Mallat's Heuristic, which says that wavelet bases are optimal for representing functions containing singularities, when there may be an arbitrary number of singularities, arbitrarily distributed.
Coorbit space theory for quasiBanach spaces
, 2005
"... We generalize the classical coorbit space theory developed by Feichtinger and Gröchenig to quasiBanach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasiBanach spaces. These atomic decompositions are used to prove fast convergence rates of be ..."
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Cited by 5 (2 self)
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We generalize the classical coorbit space theory developed by Feichtinger and Gröchenig to quasiBanach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasiBanach spaces. These atomic decompositions are used to prove fast convergence rates of best nterm approximation schemes. We apply the abstract theory to timefrequency analysis of modulation spaces M p,q m, 0 < p,q ≤ ∞.
Localization of frames II
 APPL. COMPUT. HARMON. ANAL
"... The theory of localized frames is refined to include quasiBanach spaces and spaces with multiple generators. Applications are given to nonlinear approximation with frames and to the convergence of the iterative frame algorithm in finer norms, and to the characterization of Besov spaces with wavel ..."
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Cited by 1 (1 self)
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The theory of localized frames is refined to include quasiBanach spaces and spaces with multiple generators. Applications are given to nonlinear approximation with frames and to the convergence of the iterative frame algorithm in finer norms, and to the characterization of Besov spaces with wavelet frames.