Deconvolution with unknown noise distribution is possible for multivariate signals

TL;DR

This paper demonstrates that multivariate signal deconvolution is achievable without prior knowledge of the noise distribution, using a novel estimator and model selection to adapt to unknown tail properties.

Contribution

It introduces a method for deconvolution with unknown noise distribution, establishing identifiability and providing an adaptive estimator with optimal convergence rates.

Findings

Identifiability up to translation under specific conditions.

Proposed an adaptive density estimator without noise assumptions.

Established matching lower bounds on convergence rates.

Abstract

This paper considers the deconvolution problem in the case where the target signal is multidimensional and no information is known about the noise distribution. More precisely, no assumption is made on the noise distribution and no samples are available to estimate it: the deconvolution problem is solved based only on the corrupted signal observations. We establish the identifiability of the model up to translation when the signal has a Laplace transform with an exponential growth smaller than $2$ and when it can be decomposed into two dependent components. Then, we propose an estimator of the probability density function of the signal without any assumption on the noise distribution. As this estimator depends of the lightness of the tail of the signal distribution which is usually unknown, a model selection procedure is proposed to obtain an adaptive estimator in this parameter with the same rate of convergence as the estimator with a known tail parameter. Finally, we establish a lower bound on the minimax rate of convergence that matches the upper bound.