  In this paper, we consider optimal low-rank regularized inverse matrix
approximations and their applications to inverse problems. We give an explicit
solution to a generalized rank-constrained regularized inverse approximation
problem, where the key novelties are that we allow for updates to existing
approximations and we can incorporate additional probability distribution
information. Since computing optimal regularized inverse matrices under rank
constraints can be challenging, especially for problems where matrices are
large and sparse or are only accessable via function call, we propose an
efficient rank-update approach that decomposes the problem into a sequence of
smaller rank problems. Using examples from image deblurring, we demonstrate
that more accurate solutions to inverse problems can be achieved by using
rank-updates to existing regularized inverse approximations. Furthermore, we
show the potential benefits of using optimal regularized inverse matrix updates
for solving perturbed tomographic reconstruction problems.
