# Polynomial Estimates for the Method of Cyclic Projections in Hilbert Spaces

arXiv: https://arxiv.org/abs/2205.13843 · HTML: https://mlsanity.com/paper/2205.13843/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2022.

## Built on by (most cited first, 1 of 1)

- [Continual Learning in Linear Classification on Separable Data](https://mlsanity.com/paper/2306.03534/) (2023)

## Abstract (arXiv)

We study the method of cyclic projections when applied to closed and linear subspaces Mi, 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 sumi=1mMi^perp, our result yields a polynomial rate of convergence mathcal O(k-1/2) for the method of cyclic projections itself. Moreover, if sumi=1m Mi^perp is not closed, then both of the aforementioned rates are best possible in the sense that the corresponding polynomial k1/2 cannot be replaced by k1/2+varepsilon for any varepsilon >0.