On the sharp linear convergence rate of the circumcentered--reflection method on subspaces

Abstract

For two subspaces $U,V\subseteq\RR^n$, the circumcentered--reflection method (CRM) of Behling, Bello-Cruz, and Santos~\citeyearpar{BBS2018} computes the projection onto $U\cap V$ using only the reflections across $U$ and $V$, with known linear-convergence rate equal to the cosine of the Friedrichs angle. We prove that, when CRM is initialized in $V$, it contracts at the strictly smaller rate ${\rho }_V=({\sin }^2{\theta }_p-{\sin }^2{\theta }_F)/({\sin }^2{\theta }_p+{\sin }^2{\theta }_F)$, where ${\theta }_F\in (0,\pi /2]$ is the Friedrichs angle, whose cosine we denote by $c_F:=\cos {\theta }_F$, and ${\theta }_p\in [{\theta }_F,\pi /2]$ is the largest principal angle between $U$ and $V$. The bound is sharp, attained on an explicit ray in $V$, and optimal among parameter-free single-step iterations. The constant itself is not new: Bauschke, Bello-Cruz, Nghia, Phan, and Wang~\citeyearpar{BBNPW2016} identified it as the optimal rate of the relaxed alternating-projection family and of their adaptive linesearch map $B_T$. Our contribution is that the parameter-free geometric circumcenter attains it as well, via Kantorovich's inequality applied to a single self-adjoint operator on $V$. Restricted to $V$, CRM coincides pointwise with the linesearch maps $A_T$ and $B_T$ from the Gubin--Polyak--Raik framework~\citep{GPR1967}. We further prove ${\rho }_V<c_F^2$ whenever ${\theta }_F<\pi /2$, with one-step convergence exactly when ${\theta }_F={\theta }_p$. Over-reflecting either or both of $R_U$, $R_V$ inside the circumcenter does not help. Going faster than ${\rho }_V$ universally requires memory: Chebyshev semi-iteration applied to $P_V{P}_U$ attains a strictly smaller rate, beating ${\rho }_V$ by a factor at most $2$, attained in the limit ${\theta }_F\to {\theta }_p$.

49 pages