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

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.

💡 Research Summary

The paper revisits a 1996 result of Meshulam, which states that in a finite‑dimensional Euclidean space any sequence generated by randomly applying relaxed projections onto a finite collection of affine subspaces remains bounded, even when the intersection of those subspaces is empty. Meshulam’s proof crucially relies on induction on the ambient space dimension, preventing a direct extension to infinite‑dimensional Hilbert spaces.

To overcome this obstacle, the authors introduce the notion of innate regularity for the associated family of parallel linear subspaces. A collection is innately regular if every sub‑collection is regular, i.e., the sum of the orthogonal complements of the subspaces is closed. In finite dimensions this property holds automatically, but in infinite dimensions it must be assumed.

The main technical contribution is an induction on the number ℓ of affine subspaces rather than on the dimension of the space. After recalling basic facts about orthogonal projections, relaxed projectors (R_{L,\lambda}=(1-\lambda)Id+\lambda P_L), and angle–sine–cosine relations, the authors develop several auxiliary lemmas. Proposition 3.1 shows that, under innate regularity, there exists a constant (\kappa>1) such that for any finite product of relaxed projectors the sine of the angle to the current subspace is uniformly controlled, and in particular any cycle—a product that contains each subspace at least once and at least one subspace exactly once—acts as a strict contraction on the orthogonal complement of the total subspace.

The core result, Theorem 4.2, handles a fixed relaxation parameter (\lambda\in(0,2)). For any sequence of affine subspaces ((A_n)) drawn from a finite collection (\mathcal A) and any starting point (x_0), the iterates defined by
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