Polynomial Estimates for the Method of Cyclic Projections in Hilbert Spaces

TL;DR

This paper establishes polynomial convergence rates for the method of cyclic projections in Hilbert spaces, showing optimal rates under certain conditions and analyzing the behavior of average distances to the sets involved.

Contribution

It provides new polynomial rate estimates for cyclic projections, including optimality results and behavior analysis based on the subspace sum properties.

Findings

Average distance to sets decreases as o(k^{-1/2})

Convergence rate for cyclic projections is O(k^{-1/2}) when starting from a specific subspace

Rates are proven to be optimal and cannot be improved beyond k^{1/2}

Abstract

We study the method of cyclic projections when applied to closed and linear subspaces $M_i$, $i=1,\ldots,m$, of a real Hilbert space $\mathcal H$. We show that the average distance to individual sets enjoys a polynomial behaviour $o(k^{-1/2})$ along the trajectory of the generated iterates. Surprisingly, when the starting points are chosen from the subspace $\sum_{i=1}^{m}M_i^\perp$, our result yields a polynomial rate of convergence $\mathcal O(k^{-1/2})$ for the method of cyclic projections itself. Moreover, if $\sum_{i=1}^{m} M_i^\perp$ is not closed, then both of the aforementioned rates are best possible in the sense that the corresponding polynomial $k^{1/2}$ cannot be replaced by $k^{1/2+\varepsilon}$ for any $\varepsilon >0$.