  In this work, we broadly connect kernel-based filtering (e.g. approaches such
as the bilateral filters and nonlocal means, but also many more) with general
variational formulations of Bayesian regularized least squares, and the related
concept of proximal operators. The latter set of variational/Bayesian/proximal
formulations often result in optimization problems that do not have closed-form
solutions, and therefore typically require global iterative solutions. Our main
contribution here is to establish how one can approximate the solution of the
resulting global optimization problems with use of locally adaptive filters
with specific kernels. Our results are valid for small regularization strength
but the approach is powerful enough to be useful for a wide range of
applications because we expose how to derive a "kernelized" solution to these
problems that approximates the global solution in one-shot, using only local
operations. As another side benefit in the reverse direction, given a local
data-adaptive filter constructed with a particular choice of kernel, we enable
the interpretation of such filters in the variational/Bayesian/proximal
framework.
