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Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
optimal transport; Wasserstein barycenter; concave dual; gradient ascent;
Abstract:

This paper introduces a new constraint-free concave dual formulation for the Wasserstein barycenter. Tailoring the vanilla dual gradient ascent algorithm to the Sobolev geometry, we derive a scalable Sobolev gradient ascent (SGA) algorithm to compute the barycenter for input distributions supported on a regular grid. Despite the algorithmic simplicity, we provide a global convergence analysis that achieves the same rate as the classical subgradient descent methods for minimizing nonsmooth convex functions in the Euclidean space. A central feature of our SGA algorithm is that the computationally expensive cc-concavity projection operator enforced on the Kantorovich dual potentials is unnecessary to guarantee convergence, leading to significant algorithmic and theoretical simplifications over all existing primal and dual methods for computing the exact barycenter. Our numerical experiments demonstrate the superior empirical performance of SGA over the existing optimal transport barycenter solvers.

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Overview

Overall Novelty Assessment

The paper proposes a constraint-free concave dual formulation for Wasserstein barycenters, paired with a Sobolev gradient ascent algorithm that eliminates the need for computationally expensive c-concavity projections. Within the taxonomy, it resides in the Gradient-Based Optimization Methods leaf under Computational Methods and Algorithms, alongside two sibling papers. This leaf represents a focused research direction within a broader field of fifty papers spanning theoretical foundations, computational methods, extensions, and applications, suggesting a moderately active but not overcrowded niche.

The taxonomy reveals that gradient-based methods sit within a diverse computational landscape. Neighboring leaves include Neural Network and Deep Learning Methods, which parametrize barycenters via input convex networks, and Regularization Techniques, which apply entropic or quadratic smoothing to facilitate computation. The scope note for Gradient-Based Optimization Methods explicitly excludes neural network approaches and linear programming formulations, positioning this work as a direct optimization strategy that leverages geometric structure (Sobolev spaces) rather than approximation or relaxation schemes employed in adjacent branches.

Among fourteen candidates examined across three contributions, no clearly refuting prior work was identified. The constraint-free dual formulation examined three candidates with zero refutations, the Sobolev gradient ascent algorithm examined one candidate with zero refutations, and the convergence analysis examined ten candidates with zero refutations. This limited search scope—covering top-K semantic matches and citation expansion rather than exhaustive review—suggests that within the examined literature, the combination of constraint-free dual formulation, Sobolev geometry, and convergence guarantees matching classical subgradient rates appears distinctive, though the analysis does not cover the full breadth of gradient-based optimal transport methods.

Based on the examined candidates, the work appears to occupy a relatively underexplored intersection of dual formulations, Sobolev gradient methods, and exact barycenter computation. The absence of refuting papers among fourteen candidates, combined with the sparse population of the gradient-based optimization leaf, suggests potential novelty in the specific technical approach. However, the limited search scope means this assessment reflects only a snapshot of closely related work, not a comprehensive field survey.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
14
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Computing Wasserstein barycenter of probability distributions. The field has evolved around four main branches that reflect both foundational and applied concerns. Theoretical Foundations and Properties establish the mathematical underpinnings, exploring existence, uniqueness, and regularity results such as those in Barycenters in the Wasserstein[50] and Regularity of Wasserstein barycenters[35]. Computational Methods and Algorithms form a dense branch addressing the practical challenge that Wasserstein barycenters are NP-hard[5], with approaches ranging from gradient-based optimization and entropic regularization to fixed-support schemes like Fixed-support Wasserstein barycenters[14] and scalable techniques such as Parallel Streaming Wasserstein Barycenters[15]. Extensions and Generalizations broaden the scope to conditional settings, generalized formulations as in Generalized Wasserstein Barycenters[7], and robust variants like Projection Robust Wasserstein Barycenters[45]. Applications and Domain-Specific Methods demonstrate the utility of barycenters in clustering, model ensembling, reinforcement learning, and domain adaptation, exemplified by Wasserstein Barycenter for Multi-Source[10] and Wasserstein barycenter model ensembling[18]. Within Computational Methods and Algorithms, a particularly active line of work contrasts gradient-based optimization with entropic and stochastic approaches. Gradient methods, including Sobolev Gradient Ascent for[0], leverage smooth optimization landscapes but must navigate non-convexity and high-dimensional challenges. Nearby, Fast Computation of Wasserstein[4] and Computing Wasserstein Barycenters through[11] explore alternative computational strategies that balance accuracy and scalability. The original paper Sobolev Gradient Ascent for[0] sits squarely in the gradient-based optimization cluster, emphasizing the use of Sobolev spaces to improve convergence properties compared to standard gradient methods. This contrasts with entropic regularization techniques that trade exactness for computational speed, and with stochastic methods like Stochastic Wasserstein Barycenters[13] that handle streaming or large-scale data. The interplay between theoretical guarantees, computational efficiency, and practical applicability remains a central open question across these branches.

Claimed Contributions

Constraint-free concave dual formulation for Wasserstein barycenter

The authors propose a novel unconstrained concave formulation of the Wasserstein barycenter optimization problem that achieves strong duality and fully operates in the dual space, avoiding the need for optimal transport map computations required in primal methods.

3 retrieved papers
Sobolev gradient ascent algorithm with global convergence guarantees

The authors develop a computationally simple and efficient Sobolev gradient ascent algorithm that eliminates the expensive c-concavity projection operator while retaining strong convergence guarantees, achieving the same convergence rate as classical subgradient descent methods for nonsmooth convex functions.

1 retrieved paper
Global convergence analysis matching classical subgradient methods

The authors establish a theoretical convergence rate for their SGA algorithm that matches the rate of classical subgradient descent methods in Euclidean space, providing rigorous theoretical guarantees for exact Wasserstein barycenter computation without regularization.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Constraint-free concave dual formulation for Wasserstein barycenter

The authors propose a novel unconstrained concave formulation of the Wasserstein barycenter optimization problem that achieves strong duality and fully operates in the dual space, avoiding the need for optimal transport map computations required in primal methods.

Contribution

Sobolev gradient ascent algorithm with global convergence guarantees

The authors develop a computationally simple and efficient Sobolev gradient ascent algorithm that eliminates the expensive c-concavity projection operator while retaining strong convergence guarantees, achieving the same convergence rate as classical subgradient descent methods for nonsmooth convex functions.

Contribution

Global convergence analysis matching classical subgradient methods

The authors establish a theoretical convergence rate for their SGA algorithm that matches the rate of classical subgradient descent methods in Euclidean space, providing rigorous theoretical guarantees for exact Wasserstein barycenter computation without regularization.

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