Diagnosing Forward Operator Error Using Optimal Transport
For researchers working on inverse problems with inaccurate forward models, this provides a novel diagnostic tool, though results are limited to simple cases.
The paper introduces a new tool based on the Wasserstein distance to diagnose and correct forward operator errors in overdetermined linear inverse problems. The method distinguishes noise from signals in the residual and recovers the true operator in simple cases.
We investigate overdetermined linear inverse problems for which the forward operator may not be given accurately. We introduce a new tool called the structure, based on the Wasserstein distance, and propose the use of this to diagnose and remedy forward operator error. Computing the structure turns out to use an easy calculation for a Euclidean homogeneous degree one distance, the Earth Mover's Distance, based on recently developed algorithms. The structure is proven to distinguish between noise and signals in the residual and gives a plan to help recover the true direct operator in some interesting cases. We expect to use this technique not only to diagnose the error, but also to correct it, which we do in some simple cases presented below.