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arxiv: 2605.17384 · v2 · pith:H3GXZRPLnew · submitted 2026-05-17 · 🧮 math.OC

Retractions by Alternating Projections

Shixiang Chen , Yixiao He , Wen Huang This is my paper

Pith reviewed 2026-05-21 08:30 UTC · model grok-4.3

classification 🧮 math.OC
keywords alternating projectionsretractions on manifoldsintersection of manifoldsclean intersectionsecond-order retractionmanifold optimizationNewtonSLRA
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The pith

Alternating projections between two clean-intersecting manifolds define a retraction on their common points.

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

The paper establishes that when two sufficiently smooth embedded submanifolds in Euclidean space intersect cleanly, repeated alternating projections between them converge locally to a map on the intersection manifold. This limiting map satisfies the definition of a retraction, allowing direct use in manifold-constrained optimization without explicit charts. When the manifolds have one extra degree of smoothness, the map becomes a second-order retraction. The same construction also recovers the local behavior of the NewtonSLRA method as a second-order retraction on the intersection. This supplies a new family of retraction-based algorithms for optimization problems whose feasible set is an intersection.

Core claim

Under the assumption that two C^{2,1} embedded submanifolds M1, M2 subset R^n intersect cleanly, the associated alternating mapping admits a well-defined local limiting map ψ on the intersection manifold M = M1 ∩ M2, and that ψ is a retraction on M. If, in addition, M1 and M2 are C^{3,1}, then ψ is a second-order retraction. The standard NewtonSLRA scheme can be understood as inducing a second-order retraction on M.

What carries the argument

The local limiting map ψ of the alternating projection operator, which sends nearby points to the intersection and obeys the first-order retraction conditions (identity on M and differential equal to the identity at points of M).

If this is right

  • This supplies retraction-based optimization methods for any problem whose constraint set is the intersection of two smooth manifolds.
  • The framework covers both exact and inexact alternating-projection-type iterations in a unified way.
  • NewtonSLRA on low-rank matrix problems is revealed to be a second-order retraction method on the intersection manifold.
  • Local convergence analyses for manifold optimization can now be applied directly to alternating-projection iterates on intersections.

Where Pith is reading between the lines

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

  • The construction may extend to inexact or noisy projections that still converge to the same limiting retraction.
  • Similar limiting maps could be derived for alternating projections among three or more manifolds whose total intersection is clean.
  • The retraction property opens the door to using standard Riemannian trust-region or line-search methods on intersection constraints without deriving new formulas.

Load-bearing premise

The two submanifolds must cross each other in a way that their tangent spaces together fill the whole space at every common point.

What would settle it

Take two concrete C^{2,1} surfaces in R^3 that intersect cleanly, run many steps of alternating projection from a nearby test point, and check whether the limit lies exactly on the intersection while the map's derivative at intersection points equals the identity.

Figures

Figures reproduced from arXiv: 2605.17384 by Shixiang Chen, Wen Huang, Yixiao He.

Figure 1
Figure 1. Figure 1: Tangential retraction error ∥PTR0Mr  RetrR0 (tη) − (R0 + tη)  ∥F versus the step size t in log–log scale. For very small t, the error curves plateau around 10−15 due to floating-point roundoff in double precision. The numerical results are reported in [PITH_FULL_IMAGE:figures/full_fig_p043_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Tangential retraction error ∥PTR0Mr  RetrR0 (tη) − (R0 + tη)  ∥F versus the step size t in log–log scale. For very small t, the error curves plateau around 10−15 due to floating-point roundoff in double precision. The numerical results are reported in [PITH_FULL_IMAGE:figures/full_fig_p046_1.png] view at source ↗
read the original abstract

Alternating projections and their variants are classical tools for computing points in intersections of sets. Existing analyses for smooth manifolds mainly focus on local convergence rates under transversality or related regularity conditions. In this work, we develop a unified framework for a broad class of (possibly inexact) alternating-projection-type methods on intersections of smooth manifolds. Specifically, under the assumption that two $C^{2,1}$ embedded submanifolds $\mathcal{M}_1, \mathcal{M}_2 \subset \mathbb{R}^n$ intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map $\psi$ on the intersection manifold $\mathcal{M}=\mathcal{M}_1\cap \mathcal{M}_2$, and that $\psi$ is a retraction on $\mathcal{M}$. If, in addition, $\mathcal{M}_1$ and $\mathcal{M}_2$ are $C^{3,1}$, then $\psi$ is a second-order retraction. Furthermore, the standard NewtonSLRA scheme, which exhibits quadratic local behavior under transversality, can be understood as inducing a second-order retraction on \(\M\). This framework thus provides new retraction-based optimization tools for problems constrained to the intersection manifold.

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 / 3 minor

Summary. The paper develops a unified framework showing that, for two C^{2,1} embedded submanifolds M1, M2 subset R^n that intersect cleanly, the alternating-projection mapping admits a well-defined local limiting map ψ on the intersection manifold M = M1 ∩ M2, and that ψ is a retraction on M. Under the additional assumption that M1 and M2 are C^{3,1}, ψ is a second-order retraction. The work further interprets the NewtonSLRA scheme as inducing a second-order retraction on M, thereby supplying new retraction-based tools for optimization on intersection manifolds.

Significance. If the local convergence and retraction properties hold under the stated hypotheses, the manuscript supplies a concrete bridge between classical alternating-projection methods and the retraction formalism used in manifold optimization. The clean-intersection hypothesis together with the explicit regularity classes (C^{2,1} and C^{3,1}) yields a precise setting in which the limiting map ψ can be shown to satisfy the first- and second-order retraction axioms, and the NewtonSLRA example demonstrates immediate applicability. This perspective may allow existing manifold-optimization algorithms to be applied directly to intersection-constrained problems without additional transversality assumptions.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'the associated alternating mapping' is introduced without a one-sentence definition or reference to the standard iteration; a parenthetical clarification would improve readability for readers outside the immediate area.
  2. [Main results] The transition from the C^{2,1} case (first-order retraction) to the C^{3,1} case (second-order retraction) is stated clearly, but the precise Taylor-expansion argument that upgrades the order could be highlighted with an explicit equation reference in the main theorem statement.
  3. [Framework for inexact methods] The discussion of inexact projections is mentioned in the abstract but receives limited space in the provided text; a short paragraph summarizing how the error terms are controlled in the limit would strengthen the claim of robustness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our work and for the positive assessment of its significance in bridging alternating projections with retraction-based manifold optimization. We appreciate the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under clean-intersection hypothesis

full rationale

The central result establishes existence of a local limiting map ψ from the alternating projection operator that satisfies the retraction axioms on M = M1 ∩ M2 when M1, M2 are C^{2,1} and intersect cleanly. This follows directly from standard local analysis of projections (differentiability of the projection operators near the intersection, fixed-point properties, and Taylor expansion of the composite map) once the tangent-space identity T_x M = T_x M1 ∩ T_x M2 is granted by the clean-intersection assumption. The upgrade to second-order retraction under C^{3,1} regularity is likewise a direct consequence of one extra derivative in the expansion. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation whose content is unverified outside the present work. The derivation therefore remains independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the clean-intersection assumption for C^{2,1} manifolds and the existence of the limiting map ψ; no numerical free parameters are introduced.

axioms (1)
  • domain assumption Two C^{2,1} embedded submanifolds intersect cleanly
    Invoked to ensure the alternating mapping has a well-defined local limit on the intersection manifold.
invented entities (1)
  • local limiting map ψ no independent evidence
    purpose: Defines the retraction induced by the alternating-projection iteration on the intersection
    Constructed as the limit of the alternating mapping; no independent falsifiable evidence supplied beyond the derivation itself.

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Forward citations

Cited by 1 Pith paper

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  1. Optimization over the intersection of manifolds

    math.OC 2026-05 unverdicted novelty 6.0

    Proves equivalence of clean intersection and intrinsic transversality for manifold intersections and proposes a geometric optimization method using retraction on one manifold with two orthogonal update directions and ...

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