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arxiv: 2604.15114 · v1 · submitted 2026-04-16 · 📊 stat.ML · cs.AI· cs.LG

Amortized Optimal Transport from Sliced Potentials

Minh-Phuc Truong , Khai Nguyen This is my paper

Pith reviewed 2026-05-10 09:47 UTC · model grok-4.3

classification 📊 stat.ML cs.AIcs.LG
keywords amortized optimal transportsliced optimal transportKantorovich potentialsfunctional regressiondual optimizationtransport planscolor transfermini-batch OT
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The pith

Sliced optimal transport potentials can amortize the prediction of full optimal transport plans across many measure pairs.

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

The paper develops an amortized method for optimal transport that predicts transport plans for new measure pairs by learning from simpler sliced OT potentials. It presents two strategies: regression-based amortization that fits a functional model mapping sliced potentials to full ones via least squares, and objective-based amortization that tunes the model by optimizing the Kantorovich dual. Both recover the transport plan from the resulting potentials. A reader would care because repeated OT solves are expensive in machine learning tasks like image alignment or distribution matching, and this approach reuses prior computations to approximate new ones quickly while staying independent of measure size or atom count.

Core claim

Kantorovich potentials from sliced optimal transport serve as predictors for the potentials of the full optimal transport problem. A functional regression model is fit by least-squares methods in RA-OT, or its parameters are estimated by optimizing the Kantorovich dual objective in OA-OT. The predicted OT plan is then recovered from the estimated potentials. This yields amortized solvers that reuse information across multiple measure pairs, remain parsimonious, and do not depend on structures such as the number of atoms in the discrete case.

What carries the argument

The RA-OT regression and OA-OT dual-optimization models that map sliced OT Kantorovich potentials to full OT potentials for subsequent plan recovery.

Load-bearing premise

Kantorovich potentials obtained from sliced OT contain enough information to accurately predict or optimize the full OT potentials via regression or dual objectives.

What would settle it

On held-out pairs of measures, compare the transport cost or plan recovered from the amortized potentials against the cost or plan from directly solving the full OT problem; large discrepancies would show the sliced potentials are not sufficiently informative.

Figures

Figures reproduced from arXiv: 2604.15114 by Khai Nguyen, Minh-Phuc Truong.

Figure 1
Figure 1. Figure 1: Amortized-OT uses sliced potentials to predict original dual potentials. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Wasserstein interpolations between MNIST test digits. (Top) Sinkhorn ground truth [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Prediction on a held-out spherical supply–demand instance ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Color transfer results on held-out test pairs ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Generated flow trajectories moving from a 2D Gaussian prior to the [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Flow trajectories transforming a 2D Gaussian prior to the [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualized flow trajectories mapping a 2D Gaussian prior into the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Additional qualitative visualizations of the predicted optimal transport plans on the MNIST [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Extended qualitative results on the spherical world supply-demand transport task ( [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Further qualitative examples of the color transfer task between natural images using [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Additional qualitative examples of the color transfer task ( [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
read the original abstract

We propose a novel amortized optimization method for predicting optimal transport (OT) plans across multiple pairs of measures by leveraging Kantorovich potentials derived from sliced OT. We introduce two amortization strategies: regression-based amortization (RA-OT) and objective-based amortization (OA-OT). In RA-OT, we formulate a functional regression model that treats Kantorovich potentials from the original OT problem as responses and those obtained from sliced OT as predictors, and estimate these models via least-squares methods. In OA-OT, we estimate the parameters of the functional model by optimizing the Kantorovich dual objective. In both approaches, the predicted OT plan is subsequently recovered from the estimated potentials. As amortized OT methods, both RA-OT and OA-OT enable efficient solutions to repeated OT problems across different measure pairs by reusing information learned from prior instances to rapidly approximate new solutions. Moreover, by exploiting the structure provided by sliced OT, the proposed models are more parsimonious, independent of specific structures of the measures, such as the number of atoms in the discrete case, while achieving high accuracy. We demonstrate the effectiveness of our approaches on tasks including MNIST digit transport, color transfer, supply-demand transportation on spherical data, and mini-batch OT conditional flow matching.

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 manuscript proposes two amortized methods (RA-OT and OA-OT) for predicting optimal transport plans across multiple measure pairs. RA-OT performs functional regression treating sliced-OT Kantorovich potentials as predictors and full-OT potentials as responses, estimated by least squares; OA-OT optimizes the parameters of a functional model directly on the Kantorovich dual objective. In both cases the transport plan is recovered from the estimated potentials. The methods are presented as structure-independent, parsimonious, and efficient for repeated OT problems, with empirical demonstrations on MNIST digit transport, color transfer, spherical supply-demand, and mini-batch OT conditional flow matching.

Significance. If the approximation quality holds across the claimed tasks, the work provides a practical route to amortize repeated OT computations by reusing sliced-OT information, which could reduce cost in pipelines that solve many transport problems (e.g., generative modeling or alignment tasks) while remaining independent of discrete support size.

major comments (2)
  1. [§3 (method description) and §4 (theoretical justification)] The central claim that sliced-OT potentials suffice to recover accurate full-OT plans rests on an unproven identifiability assumption. No theorem, error bound, or identifiability result is supplied showing that the averaged 1-D projections determine (or can approximate to controlled error) the multi-dimensional Kantorovich potentials; the regression or dual optimization may therefore converge to a plan whose coupling deviates substantially from the true OT plan when the measures contain couplings invisible in random projections.
  2. [Abstract and §5 (experiments)] The abstract and experimental sections state that both RA-OT and OA-OT “achieve high accuracy” on the listed tasks, yet no quantitative metrics (Wasserstein distance, relative error, runtime tables), error bars, ablation on number of slices, or direct comparison to Sinkhorn, entropic OT, or existing amortized baselines are referenced. Without these numbers it is impossible to judge whether the recovered plans are competitive or merely plausible.
minor comments (2)
  1. [§3.1] Clarify the precise functional form and parameterization of the regression model in RA-OT (e.g., is it a neural network, kernel ridge, or linear functional?) and the initialization strategy for OA-OT.
  2. [§6 (discussion)] Add a short discussion of failure modes or regimes (e.g., high-dimensional or highly anisotropic measures) where the sliced-potential approximation is expected to degrade.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment point by point below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [§3 (method description) and §4 (theoretical justification)] The central claim that sliced-OT potentials suffice to recover accurate full-OT plans rests on an unproven identifiability assumption. No theorem, error bound, or identifiability result is supplied showing that the averaged 1-D projections determine (or can approximate to controlled error) the multi-dimensional Kantorovich potentials; the regression or dual optimization may therefore converge to a plan whose coupling deviates substantially from the true OT plan when the measures contain couplings invisible in random projections.

    Authors: We appreciate the referee highlighting this theoretical gap. Our methods are motivated by the practical efficiency of sliced OT potentials as predictors for full OT potentials, building on the established approximation properties of sliced Wasserstein distances. We do not claim or prove that sliced projections universally determine the full Kantorovich potentials, nor do we supply error bounds or an identifiability theorem. The regression and optimization steps are presented as empirical amortization strategies that work well when the sliced information is sufficiently informative. In the revision we will add an explicit discussion of this limitation, clarify the assumptions, and note that a rigorous theoretical analysis of when the approximation holds (or fails) is left for future work. revision: partial

  2. Referee: [Abstract and §5 (experiments)] The abstract and experimental sections state that both RA-OT and OA-OT “achieve high accuracy” on the listed tasks, yet no quantitative metrics (Wasserstein distance, relative error, runtime tables), error bars, ablation on number of slices, or direct comparison to Sinkhorn, entropic OT, or existing amortized baselines are referenced. Without these numbers it is impossible to judge whether the recovered plans are competitive or merely plausible.

    Authors: We agree that quantitative metrics are necessary to substantiate the accuracy claims. The current experiments emphasize visual and qualitative results on MNIST transport, color transfer, spherical data, and conditional flow matching. In the revised manuscript we will expand §5 to include Wasserstein distances to ground-truth OT plans, relative errors, runtime tables, error bars from repeated runs, an ablation study on the number of slices, and direct comparisons against Sinkhorn, entropic OT, and relevant amortized baselines. These additions will be reflected in the abstract as well. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent computation of sliced vs. full potentials followed by standard fitting.

full rationale

The paper computes Kantorovich potentials from sliced OT (via 1D projections) as an independent preprocessing step, then trains either a least-squares functional regression (RA-OT) or dual-objective optimization (OA-OT) to map those to full OT potentials before recovering the plan. Training explicitly requires separate computation of both sliced and full potentials on the training measure pairs, so the fitted mapping is not tautological. No equation equates the final plan to a quantity already present in the sliced inputs by construction, no load-bearing self-citation chain is invoked, and the method is presented as an empirical amortizer whose accuracy is validated on downstream tasks rather than derived from its own definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that sliced-OT potentials contain enough information to predict full-OT potentials; no explicit free parameters, new axioms, or invented entities are named in the abstract.

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discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. ASAP: Amortized Doubly-Stochastic Attention via Sliced Dual Projection

    cs.LG 2026-05 conditional novelty 7.0

    ASAP amortizes Sinkhorn-based doubly-stochastic attention by learning a parametric map from 1D potentials to the Sinkhorn dual and reconstructing the plan via two-sided entropic c-transform, delivering 5.3x faster inf...

  2. Sliced-Regularized Optimal Transport

    stat.ML 2026-04 unverdicted novelty 7.0

    SROT regularizes the OT plan toward a smoothened sliced OT plan, producing more accurate approximations to exact OT than entropic OT while also improving on the sliced OT reference.

  3. Sliced-Regularized Optimal Transport

    stat.ML 2026-04 unverdicted novelty 7.0

    SROT regularizes the OT transport plan toward a sliced OT reference, yielding better approximations of exact OT than entropic OT and improving on the sliced OT plan itself.

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