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arxiv: 2506.07333 · v3 · submitted 2025-06-09 · 🧮 math.OC

Bregman level proximal subdifferentials and new characterizations of Bregman proximal operators

Ziyuan Wang , Andreas Themelis This is my paper

Pith reviewed 2026-05-19 11:31 UTC · model grok-4.3

classification 🧮 math.OC
keywords Bregman proximal operatorBregman level proximal subdifferentialresolventrelative smoothnessfirm nonexpansivenessasymmetryduality gapvariational analysis
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The pith

Every Bregman proximal operator equals the resolvent of a Bregman level proximal subdifferential under a standard range assumption, even without convexity.

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

The paper introduces left and right Bregman level proximal subdifferentials to overcome the limitations of classical subdifferentials when representing Bregman proximal operators in nonconvex settings. This representation holds via a standard range assumption and leads to new correspondences linking properties of the proximal operator, the underlying function, and the subdifferential. These correspondences generalize classical Euclidean results while highlighting asymmetry and duality gaps that arise naturally from the Bregman distance. The work also defines anisotropic firm nonexpansiveness and connects it to relative smoothness and convex envelopes through gradient and proximal operator properties.

Core claim

Every Bregman proximal operator turns out to be the resolvent of a Bregman level proximal subdifferential under a standard range assumption, even without convexity. Aided by this feature, new correspondences are established among useful properties of the Bregman proximal operator, the underlying function, and the left Bregman level proximal subdifferential, generalizing classical equivalences in the Euclidean case. Asymmetry and duality gap emerge as natural consequences of the Bregman distance.

What carries the argument

Bregman level proximal subdifferential (left and right versions), a subdifferential whose resolvent recovers the Bregman proximal operator under the range assumption.

If this is right

  • Properties such as firm nonexpansiveness of the proximal operator become equivalent to corresponding properties of the function and its subdifferential.
  • Existence and single-valuedness of the Bregman level proximal subdifferential can be characterized directly from the proximal operator.
  • Relative smoothness of the function is linked to coincidence results between different subdifferentials.
  • Anisotropic firm nonexpansiveness characterizes relative smooth convex functions via gradient and proximal operator behavior.

Where Pith is reading between the lines

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

  • The framework may extend subdifferential calculus tools to optimization problems that use Bregman distances but lack convexity.
  • Algorithmic schemes relying on Bregman proximal steps could be analyzed for convergence without assuming convexity of the objective.
  • The asymmetry highlighted here suggests that dual problems in Bregman settings may require separate left and right subdifferential treatments.

Load-bearing premise

The standard range assumption must hold for the resolvent representation of the Bregman proximal operator to be valid.

What would settle it

Exhibit a concrete Bregman proximal operator that satisfies the range assumption yet cannot be recovered as the resolvent of any Bregman level proximal subdifferential.

Figures

Figures reproduced from arXiv: 2506.07333 by Andreas Themelis, Ziyuan Wang.

Figure 1
Figure 1. Figure 1: Equivalences, or lack thereof, for Legendre and 1-coercive dgf [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Classic subdifferentials in variational analysis may fail to fully represent the Bregman proximal operator in the absence of convexity. In this paper, we fill this gap by introducing the left and right \emph{Bregman level proximal subdifferentials} and investigate them systematically. Every Bregman proximal operator turns out to be the resolvent of a Bregman level proximal subdifferential under a standard range assumption, even without convexity. Aided by this pleasant feature, we establish new correspondences among useful properties of the Bregman proximal operator, the underlying function, and the (left) Bregman level proximal subdifferential, generalizing classical equivalences in the Euclidean case. Unlike the classical setting, asymmetry and duality gap emerge as natural consequences of the Bregman distance. Along the way, we improve results by Kan and Song and by Wang and Bauschke on Bregman proximal operators. We also characterize the existence and single-valuedness of the Bregman level proximal subdifferential, investigate coincidence results, and make an interesting connection to relative smoothness. Abundant examples are provided to justify the necessity of our assumptions. We also introduce \emph{anisotropic firm nonexpansiveness}, a new notion that is complementary to \emph{Bregman} firm nonexpansiveness and is shown to characterize relative smooth convex functions and convex envelopes via properties of gradient and proximal operators.

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

2 major / 2 minor

Summary. The paper introduces left and right Bregman level proximal subdifferentials to represent Bregman proximal operators as resolvents under a standard range assumption, even without convexity. It derives new correspondences among properties of the operator, function, and subdifferential (generalizing Euclidean equivalences), improves results of Kan-Song and Wang-Bauschke, introduces anisotropic firm nonexpansiveness to characterize relative smoothness and convex envelopes, provides examples justifying assumptions, and connects to relative smoothness.

Significance. If the central resolvent representation holds, the work meaningfully extends Bregman proximal theory beyond convexity, supplying subdifferential characterizations and property equivalences that were previously unavailable. Credit is due for the systematic treatment of asymmetry and duality gaps induced by the Bregman distance, the improvement on prior results, the introduction of anisotropic firm nonexpansiveness, and the provision of abundant examples that test necessity of assumptions.

major comments (2)
  1. [Abstract and §1] The resolvent representation (every Bregman proximal operator equals the resolvent of a left or right Bregman level proximal subdifferential) is stated to hold under a 'standard range assumption' even without convexity. This assumption is load-bearing for the nonconvex claim, yet its precise formulation (e.g., a surjectivity or range inclusion condition on the subdifferential) and verification that it is satisfied by the proximal-operator construction itself are not made explicit. Without this, the 'even without convexity' statement remains conditional on an extra hypothesis whose automatic validity is unclear given the asymmetry of D_f.
  2. [Main representation theorem (likely §3 or §4)] In the nonconvex setting the level sets of the Bregman distance need not be closed or convex, which may affect the closedness or maximality properties required for the subdifferential to serve as a preimage under the resolvent map. The derivation of the representation should therefore include an explicit check that the range condition compensates for these failures; if the proof relies on any hidden convexity or closedness that is not justified, the central claim is at risk.
minor comments (2)
  1. [§2] Notation for left versus right versions should be introduced once and used consistently; occasional switches between 'Bregman level proximal subdifferential' and the left/right qualifiers can confuse readers.
  2. [§5] The connection to relative smoothness is interesting; a short remark clarifying how the new subdifferential recovers or differs from the classical subdifferential when f is convex would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report, which highlights both the potential significance of the work and areas where greater explicitness is needed. We address each major comment below and will revise the manuscript to strengthen the presentation of the range assumption and the nonconvex analysis.

read point-by-point responses

  1. Referee: [Abstract and §1] The resolvent representation (every Bregman proximal operator equals the resolvent of a left or right Bregman level proximal subdifferential) is stated to hold under a 'standard range assumption' even without convexity. This assumption is load-bearing for the nonconvex claim, yet its precise formulation (e.g., a surjectivity or range inclusion condition on the subdifferential) and verification that it is satisfied by the proximal-operator construction itself are not made explicit. Without this, the 'even without convexity' statement remains conditional on an extra hypothesis whose automatic validity is unclear given the asymmetry of D_f.

    Authors: We agree that the precise statement and automatic validity of the range assumption merit explicit treatment in the abstract and introduction. In the revision we will formulate the assumption as the standard range condition that the image of the (left or right) Bregman level proximal subdifferential covers the effective domain on which the proximal operator is considered; this is the natural analogue of the surjectivity condition used for resolvents of monotone operators. We will then verify directly from the definition that the subdifferential constructed from any given Bregman proximal operator satisfies this range condition by construction, without invoking convexity. The left/right distinction already incorporates the asymmetry of D_f, so the verification holds separately for each version. revision: yes

  2. Referee: [Main representation theorem (likely §3 or §4)] In the nonconvex setting the level sets of the Bregman distance need not be closed or convex, which may affect the closedness or maximality properties required for the subdifferential to serve as a preimage under the resolvent map. The derivation of the representation should therefore include an explicit check that the range condition compensates for these failures; if the proof relies on any hidden convexity or closedness that is not justified, the central claim is at risk.

    Authors: The proof of the representation theorem proceeds directly from the variational definition of the Bregman level proximal subdifferential and does not rely on convexity or closedness of Bregman level sets. The range assumption is used precisely to guarantee that, for every point in the domain, a suitable element of the subdifferential exists whose resolvent recovers the proximal operator; this bypasses the need for closed level sets or maximality in the classical sense. In the revised manuscript we will insert an explicit paragraph immediately after the statement of the theorem that spells out this compensation mechanism and confirms that no hidden convexity is employed. The examples already included in the paper further illustrate that the construction remains valid when level sets fail to be closed or convex. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces novel left and right Bregman level proximal subdifferentials as new objects to extend resolvent representations beyond convex settings. The central result—that Bregman proximal operators coincide with resolvents of these subdifferentials under a stated range assumption—is presented as a theorem derived from the definitions and properties investigated, rather than a definitional tautology or reduction to fitted parameters. Prior results by Kan-Song and Wang-Bauschke are cited only for improvement, not as load-bearing justification for the new equivalences or the asymmetry/duality observations. The derivation remains self-contained through systematic study, coincidence characterizations, relative smoothness links, and examples that test assumption necessity, without any quoted step reducing the claimed correspondences to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper relies on standard assumptions from variational analysis and Bregman geometry. It introduces new subdifferential objects by definition rather than postulating new physical entities.

axioms (1)
  • domain assumption Standard range assumption for the resolvent representation
    Explicitly invoked in the abstract as the condition under which every Bregman proximal operator is the resolvent of the new subdifferential.
invented entities (2)
  • Left and right Bregman level proximal subdifferentials no independent evidence
    purpose: To fully represent the Bregman proximal operator in the absence of convexity
    Newly defined objects whose properties are investigated systematically in the paper.
  • Anisotropic firm nonexpansiveness no independent evidence
    purpose: A new notion complementary to Bregman firm nonexpansiveness that characterizes relative smooth convex functions
    Introduced in the abstract as a novel concept shown to characterize certain function classes via gradient and proximal operators.

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Reference graph

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