Stochastic subspace correction in Hilbert space

TL;DR

This paper introduces a stochastic subspace correction method for variational problems in Hilbert spaces, demonstrating improved convergence guarantees and linking to learning algorithms in reproducing kernel Hilbert spaces.

Contribution

It presents a novel incremental approximation approach with weaker convergence conditions and explores its connection to kernel learning algorithms.

Findings

Convergence rates for expected squared error are established under weaker assumptions.

The method applies to infinite collections of subproblems in Hilbert spaces.

A link to learning algorithms in reproducing kernel Hilbert spaces is demonstrated.

Abstract

We consider an incremental approximation method for solving variational problems in infinite-dimensional Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is solved. we show that convergence rates for the expectation of the squared error can be guaranteed under weaker conditions than previously established in [Constr. Approx. 44:1 (2016), 121-139]. A connection to the theory of learning algorithms in reproducing kernel Hilbert spaces is revealed.