Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition

Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition (1992)

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by David L. Donoho

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TITLE	Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition	user correction
AUTHOR NAME	David L. Donoho	user correction
AUTHOR AFFIL	Department of Statistics; Stanford University	user correction
ABSTRACT	We describe the Wavelet-Vaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such asnumerical di erentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon Transform. We propose to solve ill-posed linear inverse problems by nonlinearly \shrinking" the WVD coe cients of the noisy, indirect data. Our approach o ers signi cant advantages over traditional SVD inversion in the case of recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L 2 loss, over the entire Besov scale. The important case of Besov spaces Bp;q, p <2, which model spatial inhomogeneity, is included. In comparison, linear procedures { SVD included { cannot attain optimal rates of convergence over such classes in the case p<2. For example, our methods achieve faster rates of convergence, for objects known to lie in the Bump Algebra or in Bounded Variation, than any linear procedure.	SVM HeaderParse 0.2
YEAR	1992	user correction
PAGES	101--126	INFERENCE
CITATIONS	44 found	ParsCit 1.0