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Barycentric Weak Inner-Product Gromov-Wasserstein

Youssef Mroueh

arXiv:2608.25145v1math.OCstat.ML

TL;DR

Pointwise GW can be too sensitive when one source state corresponds to multiple target outcomes. The paper introduces barycentric weak inner-product GW, which compares source relations through conditional means and supplies target variation by martingale gluing. It proves existence and convex-order projection results, develops ridge-based dual and algorithmic machinery, and reports PBMC prototype-to-cell gains over a scaled-identity IGW envelope.

  • Problem

    Pointwise GW comparisons can be too sensitive in one-to-many settings where several target outcomes refine one source state.

  • Method

    Barycentric wIGW compares source inner-product relations through conditional means, characterizes them via convex-order projections, and uses ridge-regularized A–B weak OT min–max optimization.

  • Results

    In PBMC prototype-to-cell transfer, wIGW exceeds the scaled-identity IGW envelope by 0.115 mean ARI and 0.060 mean NMI across five splits.

  • Takeaways & Limitations

    Mean-preserving target refinements can have zero wIGW cost, while prototype-to-cell PBMC transfer shows higher ARI and NMI for wIGW than the comparison IGW envelope.

  • Takeaways & Limitations

    The PBMC splits come from one donor, partly overlap, use six RNA-derived classes, and compare methods with unmatched solver budgets.

Abstract

from arXiv · show

Gromov-Wasserstein (GW) compares distributions through relations within each space. This pointwise comparison can be too sensitive in one-to-many settings, where several target outcomes refine one source state and their mean carries the geometry of interest. We introduce a weak GW framework that compares source relations with relations between the target conditional laws induced by a coupling. For inner-product relations, we retain the conditional means $m_π(x)=\mathbb{E}_π[Y\mid X=x]$. The resulting barycentric weak inner-product GW (wIGW) satisfies $\mathrm{wIGW}_{\mathrm{bar}}^2(μ,ν)=\inf_{η\preceq_{\mathrm{cx}}ν}\mathrm{IGW}^2(μ,η)$. Here $η\preceq_{\mathrm{cx}}ν$ means that $ν$ is a mean-preserving spread of $η$. Thus wIGW searches for an intermediate target geometry that can be refined into the prescribed target law without changing conditional means. Under finite second moments, minimizers exist and martingale gluing recovers an optimal coupling. With ridge regularization, moment duality gives an $A$-$B$ min-max problem whose inner step is weak optimal transport with a quadratic cost parameterized by $A$ and $B$; the outer problem optimizes these matrices. For finitely supported measures, we give an iterative algorithm. Under a quantitative ridge condition, the reduced problem is convex--concave, and the projected outer iteration satisfies an explicit contraction bound for inexact inner solves. Point cloud and graph feature refinement experiments illustrate how mean-preserving target refinements can have zero cost. A paired peripheral blood mononuclear cell (PBMC) multiome study evaluates atlas based cell type transfer through RNA/ATAC alignment in cell to cell and prototype to cell settings, with the prototype to cell setting representing the one-to-many case.

1 Introduction

The paper replaces pointwise target comparisons with conditional-law comparisons, focusing on conditional means for barycentric inner-product relations. It establishes convex-order, dual, algorithmic, and application results for this weak GW framework.

  • Motivation: One-to-many correspondences motivate comparing source relations with conditional target laws rather than individual target realizations.Examples include refined graph vertices, heterogeneous target cells, and coarse-to-fine quantum outcomes.
  • Barycentric formulation: Barycentric wIGW retains each conditional law only through its mean, so equal-mean conditional laws are identified.Other choices of the conditional-law relation can retain distributional spread or shape.
  • Convex-order geometry: Attainable conditional mean maps are exactly those whose pushforward lies below the target in convex order, yielding the wIGW projection onto laws η ⪯cx ν.Conditional Jensen and Strassen’s theorem provide the characterization, and the projection uses IGW over the same feasible set.
  • Existence and reconstruction: Under finite second moments, coupling, map, and projection formulations admit minimizers, and martingale gluing realizes an optimal coupling from an optimal mean map.The two-stage realization maps X to Z=m(X), then uses a martingale kernel to produce Y with target marginal ν.
  • Experiments: Mean-preserving target refinements can have zero wIGW cost, and experiments cover point clouds, graph features, and PBMC RNA/ATAC atlas-based transfer.The PBMC study evaluates both cell-to-cell and prototype-to-cell settings, with the latter representing one-to-many transfer.
  • Duality and computation: Ridge regularization produces an A–B min–max problem whose inner block is convex barycentric weak OT, with explicit reconstruction and convergence analysis under ridge conditions.The outer problem optimizes A and B; the inner cost is coercive and strongly convex in the barycentric mean map.

2 Variational foundations for IGW and weak optimal transport

This section develops variational foundations for ordinary IGW and weak optimal transport. It characterizes weak OT through conditional laws and barycenters, then connects the resulting structure to convex order, martingale couplings, and computational envelopes.

  • Ordinary IGW: Ordinary IGW becomes an outer finite-dimensional optimization over a matrix A, with each evaluation solving ordinary OT using a parametrized bilinear cost.The target marginal fixes its second moment, so only A remains variable in the envelope.
  • Weak optimal transport: Weak OT replaces pointwise target costs with costs depending on a source point and its entire conditional target distribution.The conditional law is induced by disintegrating a coupling into kernels π_x.
  • Convex order and martingales: Convex order is equivalent to the existence of a martingale coupling that recovers the less dispersed law as a conditional mean.Strassen’s characterization supplies the martingale representation used by the weak framework.
  • Barycentric formulation: Barycentric weak OT reduces the coupling problem to optimizing over conditional mean maps whose pushforward is below the target in convex order.Under uniform strong convexity, the minimizing barycentric map is unique.
  • Connection to weak GW: The relational construction compares conditional laws for source pairs, enabling weak GW to use convex-order projection and a weak OT block in later computational results.The construction replaces single conditional laws by pairs of conditional laws before aggregation against source relations.

3 A weak Gromov–Wasserstein framework based on conditional laws

This section defines weak GW by comparing source relations with relations between conditional target laws induced by a coupling. It presents two lifts and shows that ordinary GW is recovered exactly by the averaging lift.

  • Definition: Weak GW is formulated directly using the conditional laws induced by each coupling.The definition integrates a measurable conditional-law cost against the source coupling structure.
  • Two lifts: The first lift averages the original pointwise loss, whereas the second aggregates each pair of conditional laws before applying the loss.These are distinct ways to construct conditional-law relations from pointwise target relations.
  • Consistency with GW: The averaging lift provides a consistency check because disintegrating both copies of a coupling recovers the original GW objective.The exact specialization is stated as Proposition 3.2.

4 Barycentric wIGW

This section specializes weak GW to barycentric inner-product relations, retaining only conditional means. Convex order removes the coupling from the primal and establishes existence, directionality, and limits of the construction.

  • Definition: Barycentric wIGW evaluates the weak relational loss at the conditional mean map m_π.The unsquared wIGW value is the nonnegative square root of the defined objective value.
  • Barycentric reduction: Replacing pointwise target relations by conditional expectations yields a quantified reduction in the squared loss for every fixed coupling.The reduction follows from conditional Jensen’s inequality.
  • Information retained: Each conditional law enters only through its barycenter, so conditional laws with the same mean are identified.Other relations can retain more distributional information, but corresponding projection and finite A–B results require separate analysis.
  • Convex-order formulation: Convex order characterizes exactly the conditional mean maps generated by couplings with target marginal ν, thereby removing the coupling from the primal.The admissible map set is nonempty, convex, and weakly compact.
  • Directionality: The wIGW construction is directional: wIGWbar(μ, ν) and wIGWbar(ν, μ) can differ, so the authors reserve “distance” for symmetric settings.The directionality comes from the condition m#μ ⪯cx ν.

5 Moment representation and zero structure

The inner-product wIGW objective reduces to three finite-dimensional second moments of the source and conditional mean map, yielding a precise zero-cost characterization. Zero cost includes isometric embeddings refined by conditionally mean-zero target noise.

  • Moment representation: The pairwise loss reduces to the source second moment, the conditional-mean second moment, and their cross moment.These are represented by Sµ, Sm, and Mm.
  • Existence: wIGW has a minimizer because the feasible mean-map set is weakly compact and the moment objective is weakly lower semicontinuous.
  • Zero structure: wIGWbar(µ, ν) = 0 exactly when a Borel mean map satisfies the zero-set conditions in Corollary 5.2.
  • Zero structure: An isometric embedding T gives zero discrepancy whenever T#µ ⪯cx ν.The target may then contain arbitrary conditionally mean-zero noise, including heteroscedastic, anisotropic, dependent, and non-Gaussian noise.
  • Invariances: The discrepancy is invariant under separate orthogonal transformations, while centering remains necessary because raw inner products are translation-sensitive.

6 Projection onto the convex order cone

wIGW equals ordinary IGW minimized over intermediate laws below the prescribed target in convex order. This projection view ensures minimizers and enables optimal couplings through martingale gluing.

  • Projection identity: wIGW is the minimum ordinary IGW discrepancy over intermediate laws η satisfying η ⪯cx ν.
  • Monotonicity: A target mean-preserving spread cannot increase directional wIGW.
  • Comparison: Choosing η = ν directly yields wIGWbar(µ, ν) ≤ IGW(µ, ν).
  • Existence: For finite-second-moment measures, the coupling, map, and projection formulations each admit minimizers.
  • Coupling realization: An optimal mean map pushes µ to an optimal projection law and can be glued to ν by a Strassen martingale kernel.
  • Coupling realization: Optimal mean maps need not coincide when minimizers are nonunique, but either can generate an optimal wIGW coupling.

7 Duality under compact support

Under compact support, the projection formulation admits a dual representation with convex potentials and two finite-dimensional matrices. The matrices separate cross-moment and conditional-mean second-moment contributions.

  • Dual representation: Convex potentials enforce the convex-order constraint, while matrices A and B linearize the moment terms.
  • Minimax structure: The admissible mean-map set is weakly compact and convex, allowing Sion’s minimax theorem to exchange compact map minimization with potential maximization.
  • Assumptions: The compact-support dual applies to measures supported on compact sets KX and KY with finite second moments.
  • Dual representation: In the dual, u enforces convex order, A represents the source–map cross moment, and B represents the map second moment.
  • Limitation: The compact formulation preserves the outer order inf_A sup_B and does not justify exchanging the two matrix optimizations.

8 Ridge duality and the weak OT envelope

Ridge regularization converts the finite-dimensional moment representation into a coercive barycentric weak OT envelope. Under a spectral ridge condition, the outer A–B problem becomes convex–concave and supports minimax exchange.

  • Unregularized representation: The unregularized matrix reduction is exact but may lack the coercivity required for a noncompact weak OT potential dual.
  • Ridge weak OT block: A positive ridge makes the fixed-matrix weak OT cost coercive and strongly convex in the barycentric variable, yielding a unique minimizing mean map.The optimal coupling itself need not be unique.
  • Weak OT envelope: For fixed A and B, the inner problem is barycentric weak OT, with A encoding the cross moment and B encoding the conditional-mean second moment.
  • Minimax structure: If ε > 0 and ε ≥ 2λmax(Sµ), the reduced formulation is convex–concave and Fε is convex in A and concave in B.
  • Optimization: The outer problem becomes convex minimization in A after the inner weak OT block is evaluated.
  • Minimax structure: Every positive ridge yields the weak OT envelope and its potential dual, while the stronger spectral condition is needed only for outer minimax equality.

9 Equivalent formulations and computational roles

Table 2 organizes equivalent formulations of wIGW and its ridge version, clarifying how projection, moment, and weak OT representations support computation.

  • The formulations collect the projection identity, compact dual, and ridge envelope before introducing the algorithm.
  • The ridge formulation displays the weak OT oracle together with its dual over convex potentials.
  • Table 2 compares equivalent formulations of wIGW and its ridge version.

10 Exact reconstruction from the weak OT block

The weak OT block determines the optimal barycentric map, while martingale gluing reconstructs a coupling with the prescribed target marginal.

  • The weak OT envelope returns a coupling for each outer-matrix pair, while the ridge primal uses moments of its conditional mean.
  • Martingale gluing supplies the conditional variation needed to realize the target marginal after the outer optimizer is found.
  • Proposition 10.1 establishes compatibility, reconstruction, and gluing for an optimizer of the nested outer problem.
  • The reconstructed measure belongs to Π(µ, ν) and solves the ridge coupling primal.

11 Finite algorithm and convergence

For finite measures, the method combines a convex weak OT inner oracle with projected A–B updates; ridge conditions yield contraction and convergence guarantees for inexact solves.

  • Inner algorithm: Normalized KL mirror descent with Sinkhorn projections solves the fixed-matrix weak OT oracle without adding entropy to its objective.
  • Finite formulation: The finite weak OT block is a convex optimization over a transport polytope because conditional means depend linearly on the coupling matrix.
  • Inner algorithm: For fixed A and B, the oracle minimizes fA,B(P) over Π(a,b), with convexity and differentiability guaranteed by the ridge construction.
  • Finite formulation: The objective is strongly convex in the induced barycentric vector, but not necessarily in P when distinct plans share barycenters.
  • Outer algorithm: The projected outer method updates A and B using compatibility gradients, while retaining and repairing an inexact inner coupling.
  • Limitations: The implementation’s theorem does not establish an error schedule for the finite KL routine and requires stated ridge, step-size, and oracle-error hypotheses.
  • Convergence: Under the strong ridge condition, the outer problem is convex–concave and inexact inner solves produce a contraction bound with convergence for vanishing errors.
  • Convergence: The saddle operator is globally Lipschitz, with L ≤4 + 8(λX + M2(ν))/ε, and uniform inner error yields an explicit error floor.

12 Experiments

Synthetic refinements demonstrate zero weak cost for mean-preserving target expansions, while PBMC experiments evaluate transfer quality in cell-to-cell and prototype-to-cell settings.

  • Synthetic studies: The synthetic studies use point clouds and graphs to isolate zero-cost martingale refinements of Gram-preserving parent skeletons.
  • Synthetic studies: The certificate coupling assigns each target point to its generating parent, whose conditional mean recovers the clean skeleton.
  • Shape refinements: The symmetric refinement preserves every parent mean while increasing conditional spread for higher image points.
  • Shape refinements: Both shape envelope runs stopped after three iterations with zero final Frank–Wolfe gap and direct objectives agreeing with POT within 8.33 · 10−17.
  • Shape refinements: Across both noise sweeps, the weak certificate stays at arithmetic zero while ordinary IGW increases with conditional spread.
  • Graph refinement: The graph A/OT envelope stopped after two iterations with zero Frank–Wolfe gap and direct objective agreement with POT to 1.11 · 10−15.
  • PBMC transfer: In cell-to-cell transfer, wIGW exceeds the scaled-identity IGW envelope by 0.030 macro-F1, 0.082 ARI, and 0.052 NMI on average.
  • PBMC transfer: In prototype-to-cell transfer, wIGW exceeds the IGW envelope by 0.041 macro-F1, 0.115 ARI, and 0.060 NMI on average.

13 Conclusion

The conclusion positions barycentric wIGW as a conditional-mean framework for coarse-to-fine relational comparison, with convex-order structure, optimization guarantees, and empirical evidence in refinement and PBMC transfer.

  • Framework: Conditional means carry the compared geometry, while martingale kernels restore target variation and the prescribed marginal.The framework is designed for settings where conditional means encode geometry and conditional variation represents target refinement.
  • Theory: Convex-order projection identifies barycentric wIGW with IGW over intermediate target laws below the prescribed target in convex order.Under finite second moments, minimizers exist and martingale gluing realizes an optimal coupling.
  • Optimization: Ridge moment duality produces an A–B envelope with a strongly convex weak OT inner problem and contraction guarantees for projected outer iteration under stated hypotheses.The convergence result depends on spectral ridge, step-size, and oracle assumptions.
  • Empirical findings: In prototype-to-cell PBMC transfer, wIGW exceeds the IGW envelope by 0.115 mean ARI and 0.060 mean NMI on the original five splits.Both metrics are higher on all five splits, while additional same-donor splits preserve those advantages but not a uniform macro-F1 advantage.
  • Scope and limitations: The current theory and computation focus on the barycentric inner-product relation; richer conditional-law relations and infinite-dimensional or nonlinear extensions remain open.The paper names DMCov, DW2, and DMMD as richer relations whose structural, dual, and computational properties remain to be studied.

A.2.2 Proof of Theorem 2.4

This proof characterizes feasible conditional means through convex order, establishes equivalent coupling and map formulations, and derives existence, uniqueness, zero-cost, and martingale-reconstruction results.

  • Uniqueness and reconstruction: Strong convexity makes the optimal barycentric map unique, although optimal couplings realizing it may remain nonunique.Every martingale kernel from the optimal intermediate law to ν yields an optimal coupling.
  • Existence: The feasible map class is weakly compact, supporting existence of an optimal barycentric map.It is nonempty, weakly closed, and norm bounded in the reflexive Hilbert space L2(µ; R^dy).
  • Feasible maps: A coupling’s conditional mean map is feasible exactly when its pushforward is below the target law in convex order.Conditional Jensen proves necessity, while Strassen’s theorem supplies the martingale kernel for sufficiency.
  • Equivalent formulations: The weak coupling problem and its conditional-mean map problem have the same infimum.Every feasible map can be realized as the conditional mean of a feasible coupling by martingale gluing.
  • Moment structure: Finite second moments provide the moment representation and ensure the relevant objective terms are finite.The proof obtains the representation by expanding the squared loss and using independence of source copies.
  • Zero set: A feasible map preserving the source Gram relation yields zero wIGW cost, including orthogonal transformations whose pushforward lies below the target.For m(x)=Tx with T^⊤T=I, inner products are preserved exactly.

G Experimental configurations and evaluation metrics

The experimental configuration fixes finite synthetic and PBMC protocols, defines transport-based evaluation metrics, and documents robustness, split construction, and metric scope.

  • Synthetic experiments: Synthetic experiments distinguish martingale certificates, returned weak solver states, ordinary IGW comparators, and their recomputed objectives and residuals.Compatibility and marginal residuals are numerical checks rather than additional objectives.
  • PBMC data: The PBMC dataset contains paired RNA and ATAC measurements from one healthy donor, with fixed quality-control filters, six classes, and five frozen subsampling seeds.The filters retain 8212 of 12016 cells, and 7760 cells enter the eligible pool.
  • Robustness audit: The robustness audit uses fixed iteration budgets and objective-based restart selection, with reported compatibility and projection residuals treated as numerical diagnostics.The maximum objective difference under covariant controls is 2.02 · 10^-16, while residuals are not convergence certificates.
  • Evaluation protocol: PBMC comparisons use uniform empirical masses and exclude target labels and physical pair identities from transport optimization.Labels evaluate class transfer, while pair identities are reserved for the cell-to-cell retrieval diagnostic.
  • Retrieval metrics: Table 13 ranks 480 RNA cells by transported mass for each ATAC cell and reports retrieval means and sample standard deviations over five frozen splits.Random-ranking expectations are 1/480 for top-1, 5/480 for top-5, and H480/480 for mean reciprocal rank.
  • Metric scope: Retrieval is undefined for target k-means and prototype-to-cell transfer because those settings lack the required cross-modal or paired source-cell structure.Retrieval is evaluated only in the cell-to-cell task.
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