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Extending Meshulam's result on the boundedness of orbits of relaxed projections onto affine subspaces from finite to infinite-dimensional Hilbert spaces

Heinz H. Bauschke, Tran Thanh Tung

TL;DR

The paper extends Meshulam's finite-dimensional boundedness result for sequences generated by random relaxed projections onto a finite collection of affine subspaces to general Hilbert spaces under the assumption of innate regularity of the parallel linear subspaces. The core method replaces Meshulam's dimension-based induction with an induction on the number of affine subspaces and employs contraction bounds derived from cycles of relaxed projections, together with a controlled translation term when affine translates are present. Key contributions include a finite bound $C_{\\mathcal{A},\\lambda}$ ensuring boundedness for fixed and varying relaxation parameters, connections to randomized block Kaczmarz methods, and a linear-convergence result for cyclic compositions. The work also discusses the necessity of innate regularity via limiting examples and touches on extensions to polyhedral sets and parameter-sequence behavior, providing a broader theoretical framework for projection-based algorithms in infinite-dimensional settings.

Abstract

In 1996, Meshulam proved that any sequence generated in Euclidean space by randomly projecting onto affine subspaces drawn from a finite collection stays bounded even if the intersection of the subspaces is empty. His proof, which works even for relaxed projections, relies on an ingenious induction on the dimension of the Euclidean space. In this paper, we extend Meshulam's result to the general Hilbert space setting by an induction proof of the number of affine subspaces in the given collection. We require that the corresponding parallel linear subspaces are innately regular -- this assumption always holds in Euclidean space. We also discuss the sharpness of our result and make a connection to randomized block Kaczmarz methods.

Extending Meshulam's result on the boundedness of orbits of relaxed projections onto affine subspaces from finite to infinite-dimensional Hilbert spaces

TL;DR

The paper extends Meshulam's finite-dimensional boundedness result for sequences generated by random relaxed projections onto a finite collection of affine subspaces to general Hilbert spaces under the assumption of innate regularity of the parallel linear subspaces. The core method replaces Meshulam's dimension-based induction with an induction on the number of affine subspaces and employs contraction bounds derived from cycles of relaxed projections, together with a controlled translation term when affine translates are present. Key contributions include a finite bound ensuring boundedness for fixed and varying relaxation parameters, connections to randomized block Kaczmarz methods, and a linear-convergence result for cyclic compositions. The work also discusses the necessity of innate regularity via limiting examples and touches on extensions to polyhedral sets and parameter-sequence behavior, providing a broader theoretical framework for projection-based algorithms in infinite-dimensional settings.

Abstract

In 1996, Meshulam proved that any sequence generated in Euclidean space by randomly projecting onto affine subspaces drawn from a finite collection stays bounded even if the intersection of the subspaces is empty. His proof, which works even for relaxed projections, relies on an ingenious induction on the dimension of the Euclidean space. In this paper, we extend Meshulam's result to the general Hilbert space setting by an induction proof of the number of affine subspaces in the given collection. We require that the corresponding parallel linear subspaces are innately regular -- this assumption always holds in Euclidean space. We also discuss the sharpness of our result and make a connection to randomized block Kaczmarz methods.
Paper Structure (5 sections, 9 theorems, 65 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Random product of relaxed projectors
  4. Meshulam's result in infinite-dimensional spaces
  5. Applications and limiting examples

Key Result

Proposition 2.3

Let $L$ be a closed linear subspace of $X$, and let $\lambda\in\left]0,2\right[$. Then for all $x\in X$, the sine $\sin_L(x)$ and cosine $\cos_L(x)$ of the angle between $x$ and its projection $P_Lx$ are given by and Moreover, we have

Figures (1)

  • Figure 1: The first two coordinates of the relaxed random and cyclic projection sequences.

Theorems & Definitions (31)

  • proof
  • Proposition 2.3
  • proof
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6: regularity
  • Remark 2.7
  • Proposition 2.8
  • proof
  • ...and 21 more