Towards optimal sensor placement for inverse problems in spaces of measures
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This paper studies the identification of a linear combination of point sources from a finite number of measurements. Since the data are typically contaminated by Gaussian noise, a statistical framework for its recovery is considered. It relies on two main ingredients, first, a convex but non-smooth 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. Assuming standard non-degenerate source conditions as well as applying careful linearization arguments, a computable upper bound on the latter is derived. On the one hand, this allows to derive asymptotic convergence results for the mean-squared error of the estimator in the small small variance case. On the other, it paves the way for applying optimal sensor placement approaches to sparse inverse problems.
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