Unrestricted iterations of relaxed projections in Hilbert space: Regularity, absolute convergence, and statistics of displacements
C. Sinan Güntürk, Nguyen T. Thao
TL;DR
The paper analyzes unrestricted iterations of relaxed projections onto a finite family of closed subspaces in a real Hilbert space, under the innate regularity condition. It develops a geometric framework based on the angle between subspaces to quantify regularity, derives precise one-step projection formulas, and proves a global $\gamma$-moment bound: $\sum_{n=0}^\infty \|x_{n+1}-x_n\|^\gamma \le C\|x_0\|^\gamma$ for all $\gamma>0$, with a finite constant $C({\bf V},\eta,\gamma)$. As a consequence, the displacement series converges absolutely for $\gamma=1$, and the paper provides a bound on the distribution of displacement sizes, including a decay bound on the decreasing rearrangement $\delta^*_n$ of normalized displacements: $\delta^*_n< c_* e^{-\rho_* n^{1/N}}$. The results strengthen existing norm-convergence findings by giving universal moment and tail estimates for displacements, thereby clarifying the qualitative behavior of Kaczmarz-type random projections in infinite-dimensional settings and enabling quantitative control over convergence behavior.
Abstract
Given a finite collection $\mathbf{V}:=(V_1,\dots,V_N)$ of closed linear subspaces of a real Hilbert space $H$, let $P_i$ denote the orthogonal projection operator onto $V_i$ and $P_{i,λ}:= (1-λ)I + λP_i$ denote its relaxation with parameter $λ\in [0,2]$, $i=1,\dots,N$. Under a mild regularity assumption on $\mathbf{V}$ known as `innate regularity' (which, for example, is always satisfied if each $V_i$ has finite dimension or codimension), we show that all trajectories $(x_n)_{0}^\infty$ resulting from the iteration $x_{n+1} := P_{i_n,λ_n}(x_n)$, where the $i_n$ and the $λ_n$ are unrestricted other than the assumption that $\{λ_n : n \in \mathbb{N}\} \subset [η,2{-}η]$ for some $η\in (0,1]$, possess uniformly bounded displacement moments of arbitrarily small orders. In particular, we show that $$ \sum_{n=0}^\infty \|x_{n+1} - x_n \|^γ\leq C \|x_0\|^γ~\mbox{ for all }~ γ> 0,$$ where $C:=C(\mathbf{V},η,γ)<\infty$. This result strengthens prior results on norm convergence of these trajectories, known to hold under the same regularity assumption. For example, with $γ=1$, it follows that the displacements series $\sum (x_{n+1}-x_n)$ converges absolutely in $H$. Quantifying the constant $C(\mathbf{V},η,γ)$, we also derive an effective bound on the distribution function of the norms of the displacements (normalized by the norm of the initial condition) which yields a root-exponential type decay bound on their decreasing rearrangement, again uniformly for all trajectories.