# Local Kernels that Approximate Bayesian Regularization and Proximal Operators

## TL;DR

This paper demonstrates how local kernel-based filters can approximate solutions to complex Bayesian regularization problems, enabling efficient one-shot solutions and offering interpretability within a variational framework.

## Contribution

It introduces a method to approximate global Bayesian regularization solutions using locally adaptive kernels, bridging filtering techniques with variational optimization.

## Key findings

- Kernelized filters approximate global solutions for small regularization strength.
- The approach enables one-shot, local operations to solve complex optimization problems.
- Provides a framework to interpret local filters within Bayesian and variational models.

## Abstract

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.