Scalable iterative data-adaptive RKHS regularization

TL;DR

The paper introduces iDARR, a scalable iterative method for solving ill-posed inverse problems using data-adaptive RKHS regularization, which improves solution stability and accuracy over traditional norms.

Contribution

It proposes a novel generalized Golub-Kahan bidiagonalization within RKHS for efficient, data-adaptive regularization in inverse problems.

Findings

Outperforms traditional $L^2$ and $l^2$ norm regularization.

Produces stable, accurate solutions that converge as noise decreases.

Scalable with complexity $O(kmn)$ for large matrices.

Abstract

We present iDARR, a scalable iterative Data-Adaptive RKHS Regularization method, for solving ill-posed linear inverse problems. The method searches for solutions in subspaces where the true solution can be identified, with the data-adaptive RKHS penalizing the spaces of small singular values. At the core of the method is a new generalized Golub-Kahan bidiagonalization procedure that recursively constructs orthonormal bases for a sequence of RKHS-restricted Krylov subspaces. The method is scalable with a complexity of $O(kmn)$ for $m$-by-$n$ matrices with $k$ denoting the iteration numbers. Numerical tests on the Fredholm integral equation and 2D image deblurring show that it outperforms the widely used $L^2$ and $l^2$ norms, producing stable accurate solutions consistently converging when the noise level decays.