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arxiv: 2511.07953 · v2 · submitted 2025-11-11 · 🧮 math.OC

Characterization of regularity via variational stability of alternating projections sequences

Francesco Battistoni , Aris Daniilidis , Carlo Alberto De Bernardi , Enrico Miglierina This is my paper

Pith reviewed 2026-05-18 00:05 UTC · model grok-4.3

classification 🧮 math.OC
keywords regularityalternating projectionsvariational perturbationsconvex setsHilbert spacebest approximationconvergencestability
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The pith

Regularity of a pair of convex sets is equivalent to stability of alternating projections under variational perturbations.

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

The paper proves that for two nonempty closed convex sets in a Hilbert space, the geometric property of regularity is equivalent to the property that all variational perturbations of the alternating projection method still converge to a best approximation point. Earlier work had shown one direction of this equivalence under an extra boundedness assumption on the best approximation sets. The new result removes that assumption and establishes the converse, yielding a full characterization. A reader would care because regularity controls whether the classic alternating projection algorithm behaves robustly when the operator is slightly altered, which arises in many convex feasibility and optimization problems.

Core claim

A pair (A, B) of nonempty closed convex subsets of a Hilbert space is regular if and only if, for every variational perturbation of the alternating projection operator that preserves the underlying best-approximation problem, the generated sequences converge to a point in the best approximation set; this equivalence holds without any boundedness requirement on the best approximation sets.

What carries the argument

Variational perturbations of the alternating projection operator that preserve the best approximation problem for the pair (A, B).

If this is right

  • Regularity of the pair guarantees convergence of the alternating projection method under any admissible variational perturbation.
  • The stability property under perturbations serves as an equivalent characterization of regularity.
  • The equivalence holds in general Hilbert spaces and does not rely on boundedness of the best approximation sets.
  • Convergence behavior of perturbed sequences can be used to certify or refute regularity of a given pair.

Where Pith is reading between the lines

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

  • Numerical checks of convergence for a few chosen perturbations could serve as a practical test for regularity in concrete instances.
  • The same stability perspective might apply to other projection-based algorithms used in convex feasibility problems.
  • The result tightens the connection between set geometry and robustness of iterative methods without extra compactness assumptions.

Load-bearing premise

Variational perturbations act on the alternating projection operator while still preserving the underlying best approximation problem for the given pair of sets.

What would settle it

A concrete pair of closed convex sets that fails to be regular, yet for which every variational perturbation of the alternating projections still produces sequences converging to a best approximation point.

read the original abstract

The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\H$ was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair $(A,B)$ guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded.

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

1 major / 2 minor

Summary. The manuscript proves that for nonempty closed convex sets A and B in a Hilbert space, the pair (A,B) is regular (in the Borwein-Bauschke 1993 sense) if and only if every variational perturbation of the alternating projection operator yields a sequence converging to a point of the best-approximation set. The forward implication is recalled from prior work; the converse is shown by constructing a family of perturbed operators whose fixed-point behavior forces the regularity inequality to hold. The equivalence is established without any boundedness assumption on the best-approximation set, using standard weak-convergence arguments and the variational inequality characterization of projections.

Significance. If correct, the result supplies a complete characterization of regularity via variational stability of alternating projections, removing the boundedness hypothesis present in the 2022 De Bernardi-Miglierina theorem. This strengthens the theoretical foundation for convergence of projection methods in convex feasibility problems and provides a parameter-free link between a geometric property and robustness under perturbations. The proofs rely on classical tools of convex analysis and weak topology, which enhances their reliability.

major comments (1)
  1. [§4] §4, proof of Theorem 4.2 (converse): the construction of the family of variational perturbations must be checked to ensure that each perturbed operator still has the same best-approximation set as the unperturbed pair when that set is unbounded; the current argument invokes weak lower semicontinuity but does not explicitly rule out escape to infinity in the limit.
minor comments (2)
  1. [§2] Notation for the best-approximation set P_{A,B} is introduced in §2 but used without re-statement in the statement of the main theorem; a brief reminder would improve readability.
  2. [Abstract] The abstract refers to 'variational perturbations' without a one-sentence definition; adding a parenthetical reference to the 2022 definition would help readers who encounter the paper in isolation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [§4] §4, proof of Theorem 4.2 (converse): the construction of the family of variational perturbations must be checked to ensure that each perturbed operator still has the same best-approximation set as the unperturbed pair when that set is unbounded; the current argument invokes weak lower semicontinuity but does not explicitly rule out escape to infinity in the limit.

    Authors: We appreciate this observation on the proof of the converse in Theorem 4.2. The family of variational perturbations is constructed so that each perturbed operator coincides with the unperturbed alternating projection operator on the best-approximation set (by design of the perturbation term, which vanishes there), thereby preserving the set even when it is unbounded. Weak lower semicontinuity of the norm, together with the variational inequality characterization of the projections and the Fejér monotonicity inherited by the perturbed sequences, already prevents escape to infinity: any weakly convergent subsequence must land in the best-approximation set, contradicting escape. Nevertheless, to make this argument fully explicit, we will add a short clarifying paragraph immediately after the construction of the perturbations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence proved via independent constructions

full rationale

The paper recalls the 1993 definition of regularity and the 2022 definition of variational perturbations (with forward implication) from prior literature, then proves the converse direction using explicit constructions of perturbed operators, fixed-point analysis, and standard weak-convergence arguments in Hilbert space. No step reduces the main claim to a fitted parameter, self-referential definition, or unverified self-citation chain; the new result (converse without boundedness) has independent mathematical content and is externally falsifiable via the stated assumptions on closed convex sets. This is a self-contained characterization theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background from convex analysis in Hilbert spaces and on the specific definitions of regularity and variational perturbations introduced in the cited 1993 and 2022 works. No new free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • domain assumption A and B are nonempty closed convex subsets of a Hilbert space H
    This is the ambient setting stated in the abstract for the pair (A,B).
  • domain assumption Regularity of the pair (A,B) as introduced by Borwein and Bauschke (1993)
    The notion is used as the load-bearing property whose equivalence to stability is proved.

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