Risk hull method and regularization by projections of ill-posed inverse problems

Details Risk hull method and regularization by projections of ill-posed inverse problems Risk hull method and regularization by projections of ill-posed inverse problems

2006-11-08

arxiv.org/abs/math/0611228v1

Annals of Statistics 2006, Vol. 34, No. 4, 1653-1677

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000542 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000542

We study a standard method of regularization by projections of the linear inverse problem $Y=Af+\epsilon$, where $\epsilon$ is a white Gaussian noise, and $A$ is a known compact operator with singular values converging to zero with polynomial decay. The unknown function $f$ is recovered by a projection method using the singular value decomposition of $A$. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.

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