Deconvolution with unknown noise distribution is possible for multivariate signals

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.Corollary 3. Assume that there exists ρ < 2 such that R (1) and R (1) for all z 1 ∈ C d1 and z 2 ∈ C d1 , and Φ R (1) can not be identically zero since Φ R (1) (0) = 1. We then apply Theorem 1.

Deconvolution of spherical data corrupted with unknown noise

“…We first recall general results in [7]. Then, we prove a proposition which will be used to obtain the nearly parametric rate of our estimators of the radius and the center.…”

“…Then, we prove a proposition which will be used to obtain the nearly parametric rate of our estimators of the radius and the center. In [7], the authors consider the situation where the observations Y i ∈ R d come from the model…”

“…The second assumption is a mild dependence assumption (see the discussion after Theorem 2.1 in [7]).…”

“…Recently, it has been proved in [7] that deconvolution with unknown noise distribution is possible for multivariate signals, as soon as the signal can be decomposed in two components that satisfy a mild dependence assumption, that its distribution has light enough tails, and without any assumption on the noise distribution except that its two corresponding components are independently distributed. The authors of [7] then consider the situation where the probability of X 1 has a density with respect to Lebesgue measure, and they prove that not knowing the noise distribution does not deteriorate the estimation rate of the density on Sobolev regularity classes for compactly supported signals.…”

“…In Section 2, we first recall general results of [7] and we prove in Proposition 1 a strengthened version of the local L 2 -consistency of the general estimator of the characteristic function of the signal that will be a basic stone for all our convergence rates theorems. We then state our identifiability theorem, give the definition of the estimators and prove their consistency.…”