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arxiv: 2606.07888 · v1 · pith:EFHABYCQnew · submitted 2026-06-05 · 🧮 math.OC · cs.NA· math.NA

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

Yunier Bello-Cruz This is my paper

Pith reviewed 2026-06-27 20:49 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords circumcentered-reflection methodlinear convergence rateFriedrichs angleprincipal anglessubspace intersectionprojection methodsKantorovich inequalitylinesearch methods
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The pith

The circumcentered-reflection method converges at the sharp rate ρ_V when initialized in subspace V.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the circumcentered-reflection method for projecting onto the intersection of subspaces U and V achieves a linear convergence rate that depends on both the Friedrichs angle and the largest principal angle when the iteration starts inside V. This rate is strictly smaller than the previously established rate based solely on the Friedrichs angle. The result matters because it demonstrates that the parameter-free geometric method attains the same optimal rate previously identified only for certain linesearch procedures. The proof proceeds by restricting the operator to V and applying Kantorovich's inequality to a self-adjoint operator there, while also establishing pointwise coincidence with known linesearch maps.

Core claim

When initialized in V, the circumcentered-reflection method contracts at the rate ρ_V = (sin²θ_p − sin²θ_F)/(sin²θ_p + sin²θ_F), where θ_F is the Friedrichs angle and θ_p the largest principal angle between U and V. This bound is sharp, attained on an explicit ray in V, and is optimal among parameter-free single-step iterations. The constant ρ_V is derived via Kantorovich's inequality on a single self-adjoint operator on V, and the method restricted to V coincides with the linesearch maps A_T and B_T. Moreover, ρ_V is strictly less than c_F² whenever θ_F < π/2, with exact one-step convergence if θ_F = θ_p.

What carries the argument

The circumcentered-reflection operator restricted to the subspace V, equivalent to linesearch maps and analyzed using Kantorovich's inequality on one self-adjoint operator.

If this is right

  • The bound is attained along an explicit ray in V.
  • Over-reflecting either reflection inside the circumcenter does not improve the rate.
  • One-step convergence holds exactly when the Friedrichs angle equals the largest principal angle.
  • Chebyshev semi-iteration on the product of projections attains a rate strictly smaller than ρ_V, improving by a factor of at most 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Initializing in a subspace may improve convergence for other reflection-based projection methods.
  • Since faster rates require memory, hybrid methods combining single-step geometry with limited memory could be explored.
  • The optimality among single-step methods suggests that ρ_V is a natural benchmark for comparing new parameter-free algorithms on subspaces.

Load-bearing premise

The analysis relies on the circumcentered-reflection method being equivalent, when restricted to V, to the action of a single self-adjoint operator to which Kantorovich's inequality can be applied.

What would settle it

Numerically compute the norm of the iteration operator or the observed contraction ratio for CRM starting from a vector along the explicit ray in V, for subspaces with θ_F < θ_p, and verify if it matches ρ_V rather than the larger known rate.

Figures

Figures reproduced from arXiv: 2606.07888 by Yunier Bello-Cruz.

Figure 2
Figure 2. Figure 2: Remark 6.21 (Γ⋆ versus ρV is non-monotone). For any admissible (γ, β), Γ⋆ ≥ max{|1−2γ|, |1−2γ− 2β|}, and balancing these forces Γ⋆ ≥ 1/3, with the saturating choice γ ⋆ = β ⋆ = 1/3. Whenever the principal angles satisfy ρV < 1/3, equivalently b < 2a, the geometric CT outperforms the optimal linear CDR family. In the ill-conditioned regime where a/b is small, the spectrum of CT,γ,β has more flexibility and … view at source ↗
Figure 1
Figure 1. Figure 1: Theoretical contraction rates as functions of [PITH_FULL_IMAGE:figures/full_fig_p040_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Contraction rates at θp = π/2. The Chebyshev rate ρCheb is uniformly below ρV , with ratio ρV /ρCheb = (sin θp + sin θF ) 2/(sin2 θp + sin2 θF ) (Theorem 6.9). The optimal parametric CDR rate Γ⋆ is bounded below by 1/3 (Remark 6.21); for θF ≳ 0.78 the floor binds and Γ⋆ = 1/3, while ρV continues decreasing to zero at θF = θp = π/2. where a = 1/4, b = 3/4, and ρV = 1/2. In particular, (7) holds with equalit… view at source ↗
Figure 3
Figure 3. Figure 3: Decay of ∥vk−x¯0∥ for five methods, started at v0 ∈ V with ∥v0−x¯0∥ = 9.246. Asymptotic per-step rates (visible as the slopes of each curve at large k) match the theoretical predictions: DRM at cF ≈ 0.899, MAP at c 2 F ≈ 0.809, CRM at ρV ≈ 0.679, CT,γ⋆,β⋆ at Γ⋆ ≈ 0.507, and Chebyshev at ρCheb ≈ 0.392. Within the parameter-free single-step class, CT dominates; the parametric CT,γ⋆,β⋆ requires the principal … view at source ↗
Figure 4
Figure 4. Figure 4: Optimal AAMR/GAP rate of Arag´on Artacho–Campoy [1, 2] and F¨alt–Giselsson [29] [PITH_FULL_IMAGE:figures/full_fig_p047_4.png] view at source ↗
read the original abstract

For two subspaces $U,V\subseteq\RR^n$, the circumcentered--reflection method (CRM) of Behling, Bello-Cruz, and Santos~\cite{BBS2018} computes the projection onto $U\cap V$ using only the reflections across $U$ and $V$, with known linear-convergence rate $c_F$, 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 and $\theta_p\in[\theta_F,\pi/2]$ 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~\cite{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~\cite{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 $P_VP_U$ attains a strictly smaller rate, beating $\rho_V$ by a factor at most $2$, attained in the limit $\theta_F\to\theta_p$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the circumcentered-reflection method (CRM), when initialized in subspace V, converges linearly to the projection onto U ∩ V at the rate ρ_V = (sin²θ_p − sin²θ_F)/(sin²θ_p + sin²θ_F), where θ_F is the Friedrichs angle and θ_p ∈ [θ_F, π/2] is the largest principal angle between U and V. This rate is strictly smaller than the known Friedrichs rate c_F whenever θ_F < π/2, is attained on an explicit ray in V, and is optimal among parameter-free single-step iterations. The derivation proceeds by restricting CRM to V, establishing pointwise coincidence with the linesearch maps A_T and B_T, and applying Kantorovich's inequality to a single self-adjoint operator on V; additional results address over-reflection and the advantage of memory-based accelerations such as Chebyshev semi-iteration.

Significance. If the central claims hold, the work supplies a sharper, explicit convergence rate for CRM that matches the optimal rate previously identified for the relaxed alternating-projection family and the adaptive linesearch map B_T, while demonstrating that the parameter-free geometric circumcenter construction attains this rate. The independent Kantorovich-based derivation on a single operator, the explicit sharpness example, and the strict inequality ρ_V < c_F² constitute clear contributions. The comparison showing that memory is required to beat ρ_V further clarifies the method's position within the broader landscape of subspace iteration methods.

minor comments (2)
  1. [§1] §1, paragraph following Eq. (1.3): the statement that 'over-reflecting either or both of R_U, R_V inside the circumcenter does not help' would benefit from an explicit reference to the theorem or proposition that establishes this claim.
  2. The notation for the principal angles θ_p and the Friedrichs angle θ_F is introduced clearly, but a short reminder of their definitions (or a pointer to the standard reference) in the statement of the main theorem would aid readers who consult only the results section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The summary accurately captures the main results and contributions.

Circularity Check

0 steps flagged

Derivation self-contained via Kantorovich; self-citation not load-bearing

full rationale

The paper's central derivation restricts CRM to V, establishes pointwise coincidence with known linesearch maps, and obtains the rate ρ_V by applying Kantorovich's inequality to a single self-adjoint operator on V. This chain is presented as independent of the cited prior rates for other methods. The references to BBNPW2016 and BBS2018 identify the constant and the method but are not invoked to justify the CRM-specific rate; the abstract explicitly separates the contribution as the Kantorovich application. No equation reduces by construction to a fitted input or self-citation, and the sharpness claim is supported by an explicit ray construction. This is a normal minor self-citation pattern without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies exclusively on standard mathematical axioms from linear algebra and functional analysis. No free parameters are fitted, and no new entities are postulated. The contribution is in applying existing tools like Kantorovich's inequality to derive a new rate for the CRM method.

axioms (1)
  • standard math Standard properties of orthogonal projections, reflections, and principal angles between subspaces in Euclidean space R^n
    These are foundational in linear algebra and invoked implicitly throughout the analysis of convergence rates.

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