Towards optimal sensor placement for inverse problems in spaces of measures

TL;DR

This paper develops explicit, robust, and computationally feasible error estimates for sensor placement in sparse inverse problems involving measures, aiding optimal sensor configuration to improve reconstruction accuracy under noise.

Contribution

It introduces a new method to derive explicit, asymptotically sharp error bounds for sensor placement in measure-based inverse problems, incorporating stochastic noise considerations.

Findings

Derived explicit error bounds based on sensor configuration.

Provided a practical framework for guiding sensor placement.

Validated the approach with asymptotic sharpness in expectation.

Abstract

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger-Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.