Deconvolution of spherical data corrupted with unknown noise

TL;DR

This paper addresses the challenge of deconvolving spherical data with unknown noise characteristics, proposing consistent estimators for the sphere's center, radius, and signal density, with near-parametric convergence rates.

Contribution

It introduces novel estimators for spherical deconvolution with unknown noise distribution, achieving strong theoretical guarantees and optimal convergence rates.

Findings

Radius estimator has almost parametric convergence rate.

Density estimator achieves optimal rate for Sobolev classes when noise is unknown.

Estimators are consistent without additional information.

Abstract

We consider the deconvolution problem for densities supported on a $(d-1)$-dimensional sphere with unknown center and unknown radius, in the situation where the distribution of the noise is unknown and without any other observations. We propose estimators of the radius, of the center, and of the density of the signal on the sphere that are proved consistent without further information. The estimator of the radius is proved to have almost parametric convergence rate for any dimension $d$. When $d=2$, the estimator of the density is proved to achieve the same rate of convergence over Sobolev regularity classes of densities as when the noise distribution is known.