M\"{o}bius deconvolution on the hyperbolic plane with application to impedance density estimation

TL;DR

This paper introduces a novel deconvolution method on the hyperbolic plane using Helgason-Fourier calculus to estimate densities affected by random Möbius transformations, with applications in impedance reconstruction.

Contribution

It develops a minimax nonparametric density estimator on the hyperbolic plane for data corrupted by Möbius transformations, a novel approach in this context.

Findings

Provides a new statistical inverse problem framework on the hyperbolic plane.

Develops a deconvolution technique using Helgason-Fourier calculus.

Demonstrates application to impedance density estimation.

Abstract

In this paper we consider a novel statistical inverse problem on the Poincar\'{e}, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of $2\times2$ real matrices of determinant one via M\"{o}bius transformations. Our approach is based on a deconvolution technique which relies on the Helgason--Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random M\"{o}bius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincar\'{e} plane exactly describes the physical system that is of statistical interest.