Weak convergence rates for spectral regularization via sampling inequalities
Sabrina Guastavino, Gabriele Santin, Francesco Marchetti, Federico Benvenuto
TL;DR
This work develops weak convergence bounds for spectral regularization in inverse problems by replacing classical source conditions with sampling-inequality based analysis grounded in RKHS theory. It establishes deterministic bias bounds tied to the fill distance of sampling points and variance bounds under Gaussian noise, yielding weak error estimates that do not require a priori smoothness assumptions. By linking forward operators to kernel methods, the authors show that error can be controlled by geometric sampling properties, with two test-function frameworks providing robust inverse-problem guarantees. The results illuminate how kernel regularity improves rates only up to a point, and highlight practical relevance for applications with controlled sampling patterns, such as imaging and kernel-based reconstruction tasks.
Abstract
Convergence rates in spectral regularization methods quantify the approximation error in inverse problems as a function of the noise level or the number of sampling points. Classical strong convergence rate results typically rely on source conditions, which are essential for estimating the truncation error. However, in the framework of kernel approximation, the truncation error in the case of Tikhonov regularization can be characterized entirely through sampling inequalities, without invoking source conditions. In this paper, we first generalize sampling inequalities to spectral regularization, and then, by exploiting the connection between inverse problems and kernel approximation, we derive weak convergence rate bounds for inverse problems, independently of source conditions. These weak convergence rates are established and analyzed when the forward operator is compact and uniformly bounded, or the kernel operator is of trace class.