Across five frozen splits of paired RNA/ATAC data from one healthy donor, the barycentric weak transport method wIGW recorded higher ARI and NMI than the scaled-identity IGW envelope in both cell-to-cell and prototype-to-cell transfer.
The macro-F1 comparison was not uniform. In the prototype-to-cell task, target-only k-means had a mean macro-F1 of 0.590, compared with 0.577 for wIGW, although wIGW was 0.041 higher than the IGW envelope.
The method keeps a conditional average
wIGW starts with target conditional laws induced by a coupling and retains their conditional means—the average target associated with a source—for an inner-product barycentric match.
The paper characterizes the resulting projection as ordinary IGW over the same convex-order feasible set, meaning the set used for mean-preserving refinements. Under finite second moments, the coupling, map and projection formulations admit minimizers, and every feasible conditional-mean map can be realized by a two-stage coupling.
Under a stated construction, a linear isometric embedding of the source into a target that is a convex-order, mean-preserving refinement gives exactly zero barycentric wIGW. The noise example is Y = TX + ξ with E[ξ | X] = 0, meaning the added noise has zero conditional mean given X.
Ridge regularization gives the method an A–B min–max form with a convex barycentric weak-transport inner problem and a unique minimizing map for fixed matrices. In finite calculations, it uses normalized KL mirror descent with Sinkhorn scaling; entropy is the algorithm’s mirror geometry, not a penalty in the weak-transport objective.
The convergence theorem is conditional: it assumes ε > 2λX and 0 < η < 2α/L2, then gives an outer-iteration contraction factor q < 1 and bounds for inexact-oracle error.
Synthetic refinements produced near-zero weak certificates
In seeded finite-budget shape experiments, the weak certificates were 9.72 × 10⁻³⁴ for a symmetric cat refinement with n = 500 and 2.78 × 10⁻³³ for a centered Gaussian refinement with n = 250, effectively arithmetic zero. Ordinary IGW evaluated on the same refinement plans was positive, at 9.17 × 10⁻³ and 6.97 × 10⁻³. These were deterministic numerical runs.
In one graph feature instance, the weak certificate was 2.51 × 10⁻³², the weak solved primal was 9.97 × 10⁻⁶, and the ordinary refinement-plan value was 6.22 × 10⁻².
The PBMC benchmark used fixed splits
The PBMC analysis used paired RNA/ATAC data from one healthy donor. The release contained 12,016 cells; quality control retained 8,212, and 7,760 were eligible. Each of five frozen splits defined a 900-cell atlas, with 150 cells per class, and a disjoint 480-cell evaluation set, with 80 cells per class.
Cell-to-cell alignment used 480 RNA and 480 ATAC evaluation cells. Prototype-to-cell alignment used six RNA class prototypes and 480 ATAC cells. Physical pair identities and evaluation labels were excluded from the transport optimization.
Scores varied by transfer task
On cell-to-cell transfer, wIGW exceeded the scaled-identity IGW envelope by 0.030 in macro-F1, 0.082 in ARI and 0.052 in NMI. ARI and NMI were higher on all five splits.
On prototype-to-cell transfer, wIGW exceeded the IGW envelope by 0.041 in macro-F1, 0.115 in ARI and 0.060 in NMI. ARI and NMI were higher on every split, while wIGW was higher than target-only k-means by 0.083 in ARI and 0.040 in NMI.
For cell-to-cell paired retrieval, wIGW’s mean scores were 0.068 ± 0.017 for top-1, 0.203 ± 0.038 for top-5 and 0.146 ± 0.027 for mean reciprocal rank. The corresponding IGW-envelope scores were 0.056 ± 0.012, 0.064 ± 0.011 and 0.067 ± 0.012; random-ranking expectations were 0.0021, 0.0104 and 0.0141.
What the benchmark leaves open
On five fresh same-donor splits, the wIGW-minus-envelope differences were +0.008 macro-F1, +0.088 ARI and +0.063 NMI for cell-to-cell transfer. In prototype-to-cell transfer they were −0.054, +0.132 and +0.081; ARI and NMI were higher on all five new splits, while macro-F1 varied.
Ridge-sensitivity runs were stable as ε moved from 10⁻⁶ through 10⁻², declined at 10⁻¹, and only ε = 1 met the strict condition in every tested split.
The benchmark used one healthy donor, partly overlapping frozen splits, RNA-derived annotations restricted to six classes and balanced class counts that did not represent natural PBMC abundance. The split variation therefore does not measure donor-level uncertainty; reported values were means and sample standard deviations over five splits, with no inferential tests or p-values.
Method budgets were not matched by wall-clock time, and finite local solutions were sensitive to initialization and coordinate basis. All SCOT runs emitted a Sinkhorn convergence warning, with a reported maximum marginal residual of 2.52 × 10⁻⁵.
The fixed PBMC ridge value was ε = 10⁻⁴, below the strong-ridge condition used in the convergence theorem. The finite-budget runs therefore do not establish global optimization or full satisfaction of the theorem’s conditions.
Open questions include whether richer conditional-law information can remain tractable and whether these transfer findings replicate across independent donors and less balanced PBMC populations.
The manuscript is identified as arXiv:2608.25145v1 in math.OC and dated 25 August 2026.
Paper data and sources
Original title: Barycentric Weak Inner-Product Gromov-Wasserstein
Authors: Youssef Mroueh
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text