Inverse learning in Hilbert scales

Abstract

We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions the error bound can then be explicitly established.

Stochastic Gradient Descent in Hilbert Scales: Smoothness, Preconditioning and Earlier Stopping

“…for some continuous increasing function θ : R + → R + , we can give an upper bound for the distance function. Thanks to [20, Theorem 5.9], see also [35], after rescaling, we obtain…”

“…|| ||h|| H and f H satisfies a classical H ölder source condition in terms of the covariance operator T s , ensuring optimality according to [7], [5]. In [35] in the context of inverse problems, the authors derive optimality under an additional lifting condition, relating smoothness as given in terms of L −1 to smoothness in terms of T . However, it is open to show optimality, i.e.…”

“…In contrast, if the objective function has only poor smoothness properties it turns out that it is sufficient to regularize in a weaker norm to obtain optimal rates of convergence, see also [12]. A first attempt to introduce regularization in Hilbert scales in the framework of (linear) inverse learning theory has been accomplished in [35] where general spectral algorithms are investigated, but excluding SGD performed directly in an RKHS.…”