Bayesian inversion for electromyography using low-rank tensor formats

TL;DR

This paper introduces a Bayesian inversion method for electromyography that leverages low-rank tensor formats to efficiently sample from the posterior distribution, enabling high-dimensional inverse problems to be solved more feasibly.

Contribution

It develops a low-rank tensor-based approach for Bayesian inverse problems in electromyography, reducing computational effort and enabling efficient sampling.

Findings

Efficient sampling of the posterior distribution using precomputed low-rank tensor solutions.

Theoretical proof of well-posedness and low-rank representation of the forward problem.

Numerical experiments demonstrate feasibility but highlight the need for many samples for reliable estimates.

Abstract

The reconstruction of the structure of biological tissue using electromyographic data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, electromyographic data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue. Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach. Solving the discretized Bayes-inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a Metropolis-Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes' theorem proves the well-posedness of the Bayes-inverse problem. The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm. Numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters. The Metropolis-Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods.