$\ell^1$ penalty for ill-posed inverse problems

Details $\ell^1$ penalty for ill-posed inverse problems $\ell^1$ penalty for ill-posed inverse problems

2007-09-17

arxiv.org/abs/0709.2606v1

Mathematics

Statistics Theory

Scientific paper

We tackle the problem of recovering an unknown signal observed in an ill-posed inverse problem framework. More precisely, we study a procedure commonly used in numerical analysis or image deblurring: minimizing an empirical loss function balanced by an $l^1$ penalty, acting as a sparsity constraint. We prove that, by choosing a proper loss function, this estimation technique enables to build an adaptive estimator, in the sense that it converges at the optimal rate of convergence without prior knowledge of the regularity of the true solution

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