Circumcentered Reflection Method (CRM)

• CRM is a projection algorithm that computes the unique circumcenter of successive reflections to steer iterations efficiently toward feasibility.
• The method replaces traditional averaging in Douglas–Rachford with a geometric circumcenter step, achieving provably linear convergence and reduced iteration counts.
• CRM extends to multi-set and non-affine convex problems, demonstrating practical benefits in applications such as signal processing and optimization.

The circumcentered reflection method (CRM) is a projection-type algorithm introduced as an acceleration of the Douglas–Rachford method for feasibility and best-approximation problems in Hilbert spaces. CRM leverages the affine geometry of projections and reflections: rather than averaging an iterate and a doubly reflected point (as in the Douglas–Rachford scheme), CRM selects the unique circumcenter of the triangle determined by the iterate, its first reflection, and its subsequent second reflection. This geometric modification provides a more “greedy” step toward the intersection of constraint sets, and the resulting iterative process achieves linear convergence rates that are, in many cases, provably superior to classical reflection and projection methods. The method is particularly effective for affine subspaces, readily generalizes to the multi-set and non-affine convex setting, and offers improved iteration complexity both in theory and practice.

1. Geometric Principle and Iterative Scheme

CRM is fundamentally a geometric modification of the classical Douglas–Rachford method. For two affine subspaces $U, V \subset \mathbb{R}^n$ and current point $x \in \mathbb{R}^n$, the process is as follows:

• Compute the reflection $y = R_U(x) = 2P_U(x) - x$.
• Compute the double reflection $z = R_V(y) = R_V(R_U(x))$.
• Define the set $W^x = \{x, y, z\}$.
• The next iterate is set as the circumcenter of these three points:

$C_T(x) = \operatorname{circumcenter}\{x, y, z\}$

The circumcenter is uniquely defined as the point in the affine subspace $\operatorname{aff}\{x, y, z\}$ that is equidistant from all three points.

A central geometric property is that the circumcenter coincides with the projection of any point in $U \cap V$ onto $\operatorname{aff}\{x, y, z\}$:

$P_{W^x}(w) = C_T(x)\quad \forall w \in U \cap V$

Thus, CRM explicitly uses all available local reflection directions to find the "closest" step (in the plane spanned by the reflections) to the feasible set.

2. Convergence Analysis for Affine Case

The fundamental convergence result (Theorem 2.7) shows that, starting from $x$, the CRM iterates converge linearly to the best approximation $P_{U\cap V}(x)$. Let $T(x) = \frac{x + R_V R_U(x)}{2}$ be the standard Douglas–Rachford iteration. Then, for every iteration,

$$\operatorname{dist}\big(C_T(x), U \cap V\big)^2 = \operatorname{dist}\big(T(x), U \cap V\big)^2 - \|T(x) - C_T(x)\|^2$$

Hence, the improvement at each step is at least as large as Douglas–Rachford (and typically larger).

The linear convergence rate is governed by the cosine of the Friedrichs angle $c_F(U, V)$. Specifically,

$$\|C_T^k(u) - P_{U\cap V}(x)\| \leq c_F^k \|u - P_{U\cap V}(x)\|, \qquad 0 \leq c_F < 1$$

where $u \in U + V$ or any starting point projected onto $U$ or $V$. This rate is sharp for the problem class, matching the best possible rate of DRM.

3. Comparison with Douglas–Rachford and Other Methods

CRM exhibits both theoretical and practical advantages over the Douglas–Rachford method:

• Contraction per iteration: For any step, the CRM iterate is always at least as close (and typically closer) to the solution set than the DRM iterate.
• Empirical complexity: Numerical experiments indicate that iteration counts for CRM are significantly lower than DRM’s, especially under stricter convergence tolerances. For instance, all tested high-dimensional instances requiring residuals $\leq 10^{-6}$ were solved faster by CRM.
• Performance for small Friedrichs angle: When the Friedrichs angle is small, convergence of both MAP (alternating projections) and DRM deteriorates. CRM circumvents this by taking more substantial steps, yielding faster contraction.
• Non-affine case: For convex but non-affine sets (e.g., intersection of balls), CRM empirically requires fewer iterations than DRM, although proximity guarantees at each step are not universal.
• Many-set generalization: The method extends naturally to more than two sets by iteratively applying reflections across each set and computing the circumcenter of the resulting points.

Method	Iteration Map	Convergence Rate	Affine Sets Guaranteed?
Douglas–Rachford	$T(x) = \frac{x + R_V R_U(x)}{2}$	$c_F$	Yes
Alternating Proj.	$x_{k+1} = P_V P_U x_k$	$c_F^2$	Yes
Circumcentered DRM	$C_T(x) = \operatorname{circumcenter}\{x,y,z\}$	$c_F$ (and often better)	Yes

4. Extension to Multiple and Non-Affine Sets

For $m$ sets $U_1, \ldots, U_m$, a generalized CRM iteration proceeds by forming

$$\mathcal{S}^x = \left\{ x, R_{U_1}(x), R_{U_2} R_{U_1}(x), \ldots, R_{U_m} \dots R_{U_1}(x) \right\}$$

and then taking the circumcenter of this point set. Numerical evidence supports accelerated convergence even in this extended multi-set scenario, though the convergence rate and theoretical guarantees are more complicated and may depend on geometric configuration.

In the context of general convex (non-affine) feasibility, CRM has been observed to provide improved practical performance (iteration complexity)—there, however, the circumcenter may not always outdistance the classical reflection point. Preliminary theory indicates global convergence under suitable regularity assumptions.

5. Numerical Implementation and Practical Considerations

• Circumcenter computation: In the basic three-point setting, computing the circumcenter reduces to solving a $2 \times 2$ linear system corresponding to affine span and equidistance constraints. In higher dimensions or for more points (multi-set case), this increases to a system in $m$ variables but is computationally lightweight compared to eigenvalue decompositions or projections onto complex sets.
• Projection and reflection cost: As with DRM and MAP, the principal computational cost is in evaluating the projections (and, by extension, reflections). CRM does not add significant per-iteration cost unless the number of sets and ambient dimension are large.
• Initialization: For fastest convergence, the initial point may be projected onto $U$, $V$, or $U+V$.
• Stability: The method inherits the numerical conditioning of underlying projections. Existing algorithms for projections onto subspaces, balls, or halfspaces are directly applicable.
• Visualization: The method admits clear geometric intuition, as illustrated in figures provided in the original paper—where the “circumcentered” step is shown to cut directly across the “zigzag” trajectory typical of classical methods.

6. Applications and Empirical Results

CRM is suited for a variety of best approximation and feasibility problem classes:

• Best approximation in signal processing and imaging, e.g., basis pursuit, matrix completion, or compressed sensing, where the target is the closest feasible point relative to the intersection of affine or convex constraints.
• Linear and convex feasibility problems, notably in optimization, control, and systems theory, where rapid progress to feasibility is required.
• Non-affine, convex intersection problems, as in the intersection of balls, ellipsoids, or cones, where previous methods slow down considerably.

Empirical results reported in the paper (performance profiles and iteration statistics) consistently favor CRM over DRM and MAP with respect to iteration count and often with respect to CPU time—especially when the sets are nearly parallel or when high accuracy is required.

7. Theoretical and Practical Significance

CRM is mathematically significant in its explicit use of geometric information from multiple reflection steps to effect acceleration. The method:

• Achieves, and sometimes surpasses, the sharp linear convergence rate dictated by the Friedrichs angle.
• Provides per-iteration improvements over DRM as quantified by the relation

$$\operatorname{dist}(C_T(x), U \cap V)^2 = \operatorname{dist}(T(x), U \cap V)^2 - \|T(x) - C_T(x)\|^2$$

• Admits natural generalization to arbitrary numbers of (affine) sets.
• Demonstrates empirical robustness and efficiency in high-dimensional random instances.

CRM thus serves as a unifying and geometrically principled enhancement of classical projection and reflection schemes, with applicability to a broad class of feasibility and optimization problems.