Noisy Laplace deconvolution with error in the operator

TL;DR

This paper introduces an adaptive method for Laplace deconvolution in noisy settings with unknown kernels, demonstrating minimax optimality and practical effectiveness through numerical experiments.

Contribution

It presents a novel adaptive estimation procedure combining Galerkin projection and thresholding, achieving minimax optimality for noisy Laplace deconvolution with unknown kernels.

Findings

Method achieves minimax optimality under squared loss.

Procedure adapts to both signal smoothness and kernel blurring.

Numerical results show strong practical performance.

Abstract

We adress the problem of Laplace deconvolution with random noise in a regression framework. The time set is not considered to be fixed, but grows with the number of observation points. Moreover, the convolution kernel is unknown, and accessible only through experimental noise. We make use of a recent procedure of estimation which couples a Galerkin projection of the operator on Laguerre functions, with a threshold performed both on the operator and the observed signal. We establish the minimax optimality of our procedure under the squared loss error, when the smoothness of the signal is measured in a Laguerre-Sobolev sense and the kernel satisfies fair blurring assumptions. It is important to stress that the resulting process is adaptive with regard both to the target function's smoothness and to the kernel's blurring properties. We end this paper with a numerical study emphazising the good practical performances of the procedure on concrete examples.