Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition

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 as numerical differentiation, 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 coefficients of the noisy, indirect data. Our approach offers significant advantages over traditional SVD inversion in 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 L2 loss, over the entire scale of Besov spaces. The important case of Besov spaces Bσp,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.