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arxiv: 2605.06391 · v1 · submitted 2026-05-07 · 🧮 math.OC · cs.SY· eess.SY

Unbalanced Optimal Transport and Density Control for Discrete-Time Linear Systems

Haruto Nakashima , Siddhartha Ganguly , Kenji Kashima This is my paper

Pith reviewed 2026-05-08 08:09 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords unbalanced optimal transportunbalanced density controldiscrete-time linear systemsconvex optimizationGaussian measurescovariance steeringoptimal transportdensity control
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The pith

Unbalanced optimal transport and density control for discrete-time linear systems become convex when references are Gaussian.

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

This paper examines how to compare and steer probability distributions whose total mass may differ, first in a static unbalanced optimal transport setting and then in a dynamical unbalanced density control setting that respects linear dynamics and constraints. It establishes that when the references are Gaussian, both versions admit reformulations as convex optimization problems that reach global optimality, much like covariance steering does in control applications. The work includes a numerical experiment to show the method in action. A sympathetic reader would care because convexity removes the risk of suboptimal local solutions and makes the problems computationally tractable for systems where mass is created or destroyed.

Core claim

Focusing on Gaussian references and discrete-time linear systems, the unbalanced optimal transport problem and its dynamical extension to unbalanced density control both admit globally optimal convex formulations, analogous to covariance steering.

What carries the argument

The unbalanced transport functional that trades off transport cost against fidelity to reference measures, extended with linear dynamics and constraints and then specialized to Gaussians to yield a convex program.

If this is right

  • Both the static and dynamic unbalanced problems can be solved to global optimality with standard convex solvers.
  • The formulations apply directly to constrained discrete-time linear systems.
  • The approach extends the covariance steering framework to settings with unequal mass.
  • A numerical experiment confirms that the convex programs can be solved in practice.

Where Pith is reading between the lines

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

  • The same convexity might be exploitable in model-predictive control loops that must handle density creation or annihilation.
  • Feedback implementations could become feasible for real-time density steering tasks once the convex program is solved at each step.
  • Non-Gaussian references would probably force the use of approximations or relaxations to retain tractability.

Load-bearing premise

The reference measures are Gaussian and the systems are constrained discrete-time linear systems.

What would settle it

An explicit calculation for a discrete-time linear system and Gaussian references in which the unbalanced optimal transport or density control problem is shown to be non-convex.

Figures

Figures reproduced from arXiv: 2605.06391 by Haruto Nakashima, Kenji Kashima, Siddhartha Ganguly.

Figure 1
Figure 1. Figure 1: The optimal transport plans for the UOT problem with view at source ↗
Figure 4
Figure 4. Figure 4: compares the optimal transport plans for view at source ↗
read the original abstract

This article studies unbalanced optimal transport (UOT) and its dynamical extension, unbalanced density control (UDC), for a class of constrained discrete-time linear systems. UOT compares measures with unequal total mass by balancing transport cost and fidelity to reference measures, while UDC incorporates system dynamics and constraints into this framework. Focusing on Gaussian references and discrete-time linear systems, we show that both problems admit globally optimal convex formulations, analogous to covariance steering. A numerical experiment is provided to illustrate our approach.

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

0 major / 2 minor

Summary. The manuscript studies unbalanced optimal transport (UOT) and its dynamical extension, unbalanced density control (UDC), for constrained discrete-time linear systems. Focusing on Gaussian reference measures, it claims that both problems admit globally optimal convex formulations analogous to covariance steering and illustrates the approach with a numerical experiment.

Significance. If the convexity and global optimality claims hold under the stated Gaussian and linear-system assumptions, the work extends the covariance-steering framework to unbalanced settings where total mass need not be conserved. This could enable efficient convex optimization for density-control problems in linear dynamics, with the numerical experiment serving as initial validation of practical applicability.

minor comments (2)
  1. The abstract states that a numerical experiment is provided but supplies no details on the system matrices, constraints, reference Gaussians, or quantitative results; this should be expanded in the main text or a dedicated section to allow assessment of the claimed convexity benefits.
  2. The analogy to covariance steering is invoked repeatedly; a brief recap of the key covariance-steering convex program (with citation) in the introduction would clarify the precise extension being made.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review of our manuscript on unbalanced optimal transport and unbalanced density control for discrete-time linear systems. The referee's summary accurately captures the scope and claims of the work, and we appreciate the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper scopes its claims to Gaussian reference measures and constrained discrete-time linear systems, asserting that UOT and UDC then admit globally optimal convex formulations analogous to covariance steering. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or self-citation chains. The abstract and problem formulation present independent content within the stated regime, with no evidence of renaming known results or smuggling ansatzes via prior self-work. The derivation chain is self-contained against external benchmarks for the scoped class.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from linear control theory and optimal transport; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The systems are constrained discrete-time linear systems
    Explicitly stated as the class of systems considered.
  • domain assumption References are Gaussian
    The focus is on Gaussian references for which convexity holds.

pith-pipeline@v0.9.0 · 5380 in / 1236 out tokens · 67734 ms · 2026-05-08T08:09:59.337125+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

  1. [1]

    Foundations of Computational Mathematics , year=

    Gabriel Peyré and Bernhard Schmitzer and François-Xavier Vialard , title=. Foundations of Computational Mathematics , year=

    work page

  2. [2]

    Moslehian and Mohsen Kian and Qingxiang Xu , title=

    Mohammad S. Moslehian and Mohsen Kian and Qingxiang Xu , title=. Banach Journal of Mathematical Analysis , year=

    work page

  3. [3]

    Balci and Efstathios Bakolas , title=

    Isin M. Balci and Efstathios Bakolas , title=. 2022 , eprint=

    work page 2022

  4. [4]

    Montesuma and Fred N

    Eduardo F. Montesuma and Fred N. Mboula and Antoine Souloumiac , title=. IEEE Transactions on Pattern Analysis and Machine Intelligence , year=

    work page

  5. [5]

    IMA Journal of Applied Mathematics , year=

    Louis-Philippe Saumier and Martial Agueh and Boualem Khouider , title=. IMA Journal of Applied Mathematics , year=

    work page

  6. [6]

    Handbook of Numerical Analysis , year=

    Thibault Séjourné and Gabriel Peyré and François-Xavier Vialard , title=. Handbook of Numerical Analysis , year=

    work page

  7. [7]

    Zavlanos and and Karl H

    Zifan Wang and Yi Shen and Michael M. Zavlanos and and Karl H. Johansson , title=. Advances in Neural Information Processing Systems , year=

    work page

  8. [8]

    IEEE Control Systems Letters , year=

    Kohei Morimoto and Kenji Kashima , title=. IEEE Control Systems Letters , year=

    work page

  9. [9]

    Transactions on Automatic Control , year=

    Fengjiao Liu and George Rapakoulias and Panagiotis Tsiotras , title=. Transactions on Automatic Control , year=

    work page

  10. [10]

    SIAM Review , volume=

    Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schrodinger bridge , author=. SIAM Review , volume=. 2021 , publisher=

    work page 2021

  11. [11]

    Data-driven

    Haruto Nakashima and Siddhartha Ganguly and Kenji Kashima , booktitle=. Data-driven. 2025 , volume=

    work page 2025

  12. [12]

    arXiv, ://arxiv.org/abs/2512.06797, arXiv:2512.06797 [math], doi:10.48550/arXiv.2512.06797

    Optimal and Diffusion Transports in Machine Learning , author=. arXiv preprint arXiv:2512.06797 , year=

    work page internal anchor Pith review arXiv

  13. [13]

    2026 , eprint=

    Globally Solving Unbalanced Optimal Transport and Density Control for Gaussian Distributions , author=. 2026 , eprint=

    work page 2026