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arxiv: 2606.27767 · v1 · pith:64VZZKRAnew · submitted 2026-06-26 · 💻 cs.LG · math.OC

Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization

Cl\'ement Bonet , Pierre-Cyril Aubin-Frankowski , Youssef Mroueh This is my paper

Pith reviewed 2026-06-29 05:00 UTC · model grok-4.3

classification 💻 cs.LG math.OC
keywords difference of convex programmingWasserstein spaceMMD optimizationconvex-concave procedureprobability measuresnon-convex optimizationenergy distance
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The pith

A lifted convex-concave procedure finds almost-stationary points for difference-of-convex objectives over the Wasserstein space.

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

The paper develops a way to optimize non-convex functionals defined on probability measures by first decomposing them as differences of convex functions and then lifting the classical convex-concave procedure to the Wasserstein geometry. Under smoothness and strong convexity of the convex parts, the resulting iterates are shown to be almost stationary. Explicit decompositions are constructed for the maximum mean discrepancy and energy distance, and local convergence is established under mild conditions. The work matters because many distribution-matching objectives used in machine learning are non-convex along Wasserstein geodesics, where ordinary first-order methods lack strong guarantees.

Core claim

Objectives over the Wasserstein space that admit a difference-of-convex decomposition can be minimized by a lifted CCCP algorithm whose iterates are almost stationary when the convex components are smooth and strongly convex. Explicit Wasserstein DC decompositions are supplied for the MMD and energy-distance functionals, and the scheme is shown to converge locally under mild assumptions. Well-chosen decompositions produce faster and more stable convergence than plain Wasserstein gradient descent on these objectives.

What carries the argument

The lifted convex-concave procedure (CCCP) applied to a DC decomposition of a functional on the Wasserstein space, which generates a sequence of convex subproblems solved along Wasserstein geodesics.

If this is right

  • Iterates of the lifted CCCP are almost stationary for smooth strongly convex DC components.
  • Explicit DC decompositions exist for both MMD and energy distance.
  • Local convergence holds under mild assumptions on those decompositions.
  • Well-chosen DC decompositions converge faster and more stably than Wasserstein gradient descent on MMD objectives.

Where Pith is reading between the lines

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

  • The approach may extend to other optimal-transport discrepancies once suitable DC decompositions are found.
  • It could be combined with existing Wasserstein gradient flows to handle mixed convex-nonconvex objectives in generative modeling.
  • Finding systematic ways to obtain DC decompositions for arbitrary functionals on measures remains a separate algorithmic question.

Load-bearing premise

The convex parts of the DC decomposition must be smooth and strongly convex.

What would settle it

Run the lifted CCCP on an MMD objective whose DC decomposition has a convex part that is not strongly convex and check whether the iterates remain far from stationarity.

Figures

Figures reproduced from arXiv: 2606.27767 by Cl\'ement Bonet, Pierre-Cyril Aubin-Frankowski, Youssef Mroueh.

Figure 2
Figure 2. Figure 2: Convergence of WGD and WC￾CCP to minimize F(µ) = 1 2 ED(µ, ν) with ν samples from CIFAR10. Proposition 8. Let q ∈ C 2 ([0, S∗]). Assume there exists q± ∈ C 2 ([0, S∗]) such that q = q+ − q−. If λ[q+], λ[q−] ≥ 0, then the translation-invariant kernel k(x, y) = ψ(x − y) = q(∥x − y∥ 2 2 ) satisfies Proposition 7 with ψ±(z) = q±(∥z∥ 2 2 ), α ± = λ[q±]. Moreover, if Λ[q+],Λ[q−] < ∞, for all k ≥ 0, the Lipschitz… view at source ↗
Figure 3
Figure 3. Figure 3: Optimization of F(µ) = 1 2MMD2 k (µ, ν) for ν a Gaussian target and k the Gaussian kernel. (Left) Evolution of the squared MMD along the flow. (Right) Trajectories of the particles over time. The initial particles are in blue and the final particles in red. comparing the three schemes in the same setting, assuming access to an oracle for WCCCP and FB. We add in Figure E.7 a comparison with different step s… view at source ↗
read the original abstract

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.

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 / 1 minor

Summary. The manuscript proposes lifting the classical convex-concave procedure (CCCP) to the Wasserstein space for optimizing functionals that admit a difference-of-convex (DC) decomposition. Under smoothness and strong-convexity assumptions on the convex summands, it claims to prove that the resulting algorithm produces almost-stationary iterates. Explicit DC decompositions are developed for the Maximum Mean Discrepancy (MMD) and Energy Distance (ED) functionals, for which local convergence is established under mild assumptions. Empirical comparisons indicate that suitably chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on MMD objectives.

Significance. If the smoothness and strong-convexity assumptions hold for the explicit DC splittings of MMD and ED with respect to the Wasserstein metric, the work supplies a new first-order scheme for a practically relevant class of non-convex problems on probability measures together with concrete constructions and local-convergence guarantees. The provision of explicit DC decompositions for MMD and ED constitutes a concrete, reusable contribution.

major comments (1)
  1. [Abstract] Abstract: the almost-stationarity and local-convergence statements are conditioned on the convex components of the DC decomposition being smooth and strongly convex in the Wasserstein geometry. The manuscript supplies explicit DC decompositions for MMD and ED yet provides no verification, constant bounds, or counter-example checks confirming that these particular summands satisfy strong convexity (or even finite smoothness constants) with respect to the Wasserstein metric; this verification is load-bearing for the applicability of the theoretical guarantees to the functionals highlighted in the title and abstract.
minor comments (1)
  1. [Abstract] The abstract refers to 'well-chosen DC decompositions' without stating the selection criterion or the precise mild assumptions used for local convergence of MMD/ED.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the abstract. We address the point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the almost-stationarity and local-convergence statements are conditioned on the convex components of the DC decomposition being smooth and strongly convex in the Wasserstein geometry. The manuscript supplies explicit DC decompositions for MMD and ED yet provides no verification, constant bounds, or counter-example checks confirming that these particular summands satisfy strong convexity (or even finite smoothness constants) with respect to the Wasserstein metric; this verification is load-bearing for the applicability of the theoretical guarantees to the functionals highlighted in the title and abstract.

    Authors: We agree that the general almost-stationarity result relies on smoothness and strong-convexity of the convex summands with respect to the Wasserstein metric, and that the manuscript does not supply explicit verification, bounds, or checks for the specific DC components of MMD and ED. The local-convergence claim for MMD/ED is stated under separate mild assumptions, but the referee is correct that the abstract does not clearly separate the general theorem from the application. In the revision we will add a dedicated paragraph (or short appendix) that either derives the relevant constants for the chosen decompositions or states the precise additional conditions needed for the MMD/ED cases. This addresses the load-bearing gap without altering the main theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence claims rest on independent assumptions and explicit decompositions without reduction to self-defined inputs.

full rationale

The abstract states that almost-stationarity is proved under smoothness and strong-convexity assumptions on the DC components, with explicit decompositions supplied for MMD and ED and local convergence shown under mild assumptions. No quoted equations or steps reduce any prediction or theorem to a fitted parameter, self-citation chain, or definitional equivalence. The derivation chain is presented as conditional on external assumptions that are not shown to be tautological with the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of a DC decomposition whose convex pieces satisfy smoothness and strong convexity; these are domain assumptions imported from convex analysis rather than derived in the paper.

axioms (1)
  • domain assumption The objective functional admits a DC decomposition whose convex parts are smooth and strongly convex along Wasserstein geodesics.
    Stated in the abstract as the condition under which almost stationarity and local convergence hold.

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