Towards optimal sensor placement for inverse problems in spaces of measures
arxiv(2023)
摘要
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.
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