Fisher Information for Inverse Problems and Trace Class Operators

TL;DR

This paper develops a mathematical framework for analyzing Fisher information in infinite-dimensional inverse problems with Gaussian noise, focusing on trace class covariance operators and the Cameron-Martin space, with applications to electromagnetic source problems.

Contribution

It introduces conditions for the Fisher information to be well-defined and trace class in infinite-dimensional inverse problems, linking Jacobian and covariance operator properties.

Findings

Fisher information is well-defined in the Cameron-Martin space.

Conditions relate Jacobian singular values and covariance eigenvalues.

Explicit example with electromagnetic inverse source problem.

Abstract

This paper provides a mathematical framework for Fisher information analysis for inverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, and the Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriate space for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictly smaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information to be trace class, a sufficient condition is formulated based on the singular values of the Jacobian as well as of the eigenvalues of the covariance operator, together with some regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with "external" spherically isotropic noise, as well as "internal" additive uncorrelated noise.