Minimax estimation via wavelet shrinkage

Abstract

We attempt to recover an unknown function from noisy, sampled data.\nUsing orthonormal bases of compactly supported wavelets, we develop a nonlinear\nmethod which works in the wavelet domain by simple nonlinear shrinkage of the\nempirical wavelet coefficients. The shrinkage can be tuned to be nearly minimax\nover any member of a wide range of Triebel- and Besov-type smoothness\nconstraints and asymptotically mini-max over Besov bodies with $p \\leq q$.\nLinear estimates cannot achieve even the minimax rates over Triebel and Besov\nclasses with $p<2$, so the method can significantly outperform every linear\nmethod (e.g., kernel, smoothing spline, sieve in a minimax sense). Variants of\nour method based on simple threshold nonlinear estimators are nearly minimax.\nOur method possesses the interpretation of spatial adaptivity; it\nreconstructs using a kernel which may vary in shape and bandwidth from point to\npoint, depending on the data. Least favorable distributions for certain of the\nTriebel and Besov scales generate objects with sparse wavelet transforms. Many\nreal objects have similarly sparse transforms, which suggests that these\nminimax results are relevant for practical problems. Sequels to this paper,\nwhich was first drafted in November 1990, discuss practical implementation,\nspatial adaptation properties, universal near minimaxity and applications to\ninverse problems.

Keywords

Affiliated Institutions

• Stanford University US

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