arxiv: 2605.11429 · v1 · submitted 2026-05-12 · 🧮 math.OC

Recognition: 2 theorem links

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From Schrodinger Bridge to Optimal Transport over Sub-Riemannian Manifolds

Bahman Gharesifard, Daniel Owusu Adu, Karthik Elamvazhuthi

Pith reviewed 2026-05-13 02:30 UTC · model grok-4.3

classification 🧮 math.OC

keywords Schrödinger bridgeoptimal transportsub-Riemannian manifoldentropic regularizationdegenerate diffusionSinkhorn algorithm

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The pith

Entropic regularization via Schrödinger bridge enables numerical solution of optimal transport on sub-Riemannian manifolds and recovers the deterministic case as noise vanishes.

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

The authors examine how to reshape probability distributions under constraints of sub-Riemannian geometry, which arises in underactuated systems. They introduce small noise along the allowed directions to create an entropic regularization of the optimal transport problem, turning it into a Schrödinger bridge for a degenerate diffusion. Assuming the vector fields satisfy bracket-generating conditions, the transition densities become smooth and strictly positive. This setup admits a forward-backward characterization of the optimal bridge, which in turn supports a Sinkhorn-type algorithm to compute the necessary potentials. In the limit of vanishing noise, the construction recovers the original deterministic sub-Riemannian optimal transport problem, as demonstrated by a numerical example.

Core claim

We study the least-energy way to reshape a probability distribution when motion is constrained to a horizontal bundle. To obtain a continuous and numerically tractable formulation, we introduce an entropic regularization by adding small noise aligned with the control directions and study the associated Schrödinger bridge problem. The resulting reference process is a degenerate diffusion on the sub-Riemannian manifold. Under bracket-generating hypotheses we obtain smooth, strictly positive transition densities and a forward-backward characterization of the optimal bridge. This leads to a practical Sinkhorn-type algorithm for the Schrödinger potentials and, as the noise level vanishes, a 0.0pt

What carries the argument

The Schrödinger bridge for a degenerate diffusion on a sub-Riemannian manifold obtained by adding noise along horizontal control directions, which regularizes the optimal transport problem.

If this is right

Smooth strictly positive transition densities for the reference process under bracket-generating hypotheses.
Forward-backward characterization of the optimal Schrödinger bridge.
Practical Sinkhorn-type algorithm to compute the Schrödinger potentials.
Recovery of the deterministic sub-Riemannian optimal transport problem in the vanishing-noise limit.

Where Pith is reading between the lines

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

The construction could extend density-control methods to a wider class of nonholonomic mechanical systems.
Similar noise-based regularizations may render other degenerate geometric transport problems numerically tractable.
The forward-backward structure opens the door to iterative schemes on manifolds where direct dynamic programming is intractable.

Load-bearing premise

The bracket-generating hypotheses on the sub-Riemannian manifold must hold to ensure the degenerate diffusion has smooth and strictly positive transition densities.

What would settle it

A computation on the Heisenberg group showing that the transition densities are not smooth and positive despite the bracket-generating condition, or that the Sinkhorn algorithm fails to recover known deterministic optimal transport costs as noise approaches zero.

Figures

Figures reproduced from arXiv: 2605.11429 by Bahman Gharesifard, Daniel Owusu Adu, Karthik Elamvazhuthi.

**Figure 1.** Figure 1: Plots (a) and (b) are anisotropic diffusion dXt = √ ϵg(Xt)dWt in R 3 with ϵ = 1 as opposed to (c) which is an isotropic diffusion dXt = √ ϵdWt with ϵ = 1 in R 3 . In all scenarios, the same 10 initial samples from X0 ∼ µ0 are used. The plot in (a) is a horizontally constrained (hypoelliptic) diffusion of Heisenberg type, encoded by g in Example 5.1, where the noise acts only along the horizontal distributi… view at source ↗

**Figure 2.** Figure 2: The above is the plot of the isosurfaces of the initial and final densities defined in (5.2) and (5.3), respectively. Once the fixed point φf is obtained, the full space-time solution for (3.10a)-(3.10c) is recovered by φ(t, x) = (Qtf −tφf )(x), and φb(t, x) = (Pt [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: The plots above are the isosurface plots of the optimal density ρ(t) associated to Example 5.1, with α = 1 4 . The aim is to demonstrate a smooth and continuous morphing from the initial Gaussian blob in (5.2) to the final ring-shaped distribution in (5.3), for any ϵ > 0. Hence, as ϵ → 0, the Schrödinger bridge approaches deterministic optimal transport. bridge problem, interpreted as the optimal change in… view at source ↗

**Figure 4.** Figure 4: This plot is the corresponding Heisenberg bridge path associated to Example 5.1 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

read the original abstract

We study the least-energy way to reshape a probability distribution when motion is constrained to a horizontal bundle, that is, optimal transport and distribution steering in sub-Riemannian geometry, motivated by density control over underactuated systems. To obtain a continuous and numerically tractable formulation, we introduce an entropic regularization by adding small noise aligned with the control directions and study the associated Schrodinger bridge problem. The resulting reference process is a degenerate diffusion on the sub-Riemannian manifold. Under bracket-generating hypotheses we obtain smooth, strictly positive transition densities and a forward--backward characterization of the optimal bridge. This leads to a practical Sinkhorn-type algorithm for the Schrodinger potentials and, as the noise level vanishes, a recovery of the deterministic sub-Riemannian optimal transport problem. We demonstrate with a numerical example.

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.

Desk Editor's Note private letter to a colleague

The paper sets up a Schrödinger bridge on sub-Riemannian manifolds to regularize optimal transport for underactuated systems and gives a Sinkhorn scheme that recovers the deterministic case.

read the letter

The core contribution is a workable entropic regularization for optimal transport when motion is restricted to a horizontal distribution. By adding noise only along the control directions they turn the problem into a Schrödinger bridge on the sub-Riemannian manifold, then exploit the bracket-generating condition to guarantee smooth positive transition densities. From there the forward-backward characterization follows, the potentials are computed with a Sinkhorn iteration, and the zero-noise limit recovers the original deterministic sub-Riemannian transport problem. The numerical example illustrates that the scheme runs in practice. This is a direct and useful adaptation of existing entropic OT tools to a geometric control setting that had not been treated this way before. The argument structure is standard once the hypoellipticity step is in place, and the stress-test note confirms there is no internal gap or circularity. The only real scope limitation is the bracket-generating hypothesis itself; without it the densities need not be positive and the algorithm does not apply, but that is an honest restriction rather than a hidden flaw. The paper is aimed at researchers in geometric control and stochastic processes on manifolds who need a computable proxy for distribution steering under nonholonomic constraints. A reader already comfortable with sub-Riemannian geometry and entropic regularization will get immediate value from the algorithm and the limit result. It is solid enough and novel enough to deserve a serious referee, even if the proofs will need careful checking in review.

Referee Report

2 major / 3 minor

Summary. The manuscript extends the Schrödinger bridge and entropic optimal transport to sub-Riemannian manifolds by regularizing the deterministic horizontal control problem with small noise in the control directions, yielding a degenerate diffusion as reference process. Under bracket-generating hypotheses it establishes smooth strictly positive transition densities, derives a forward-backward characterization of the optimal bridge, obtains a practical Sinkhorn-type algorithm for the Schrödinger potentials, and proves that the zero-noise limit recovers the deterministic sub-Riemannian optimal transport problem; the claims are illustrated by a numerical example.

Significance. If the derivations hold, the work supplies a principled, numerically tractable route to distribution steering on underactuated systems and bridges stochastic and deterministic sub-Riemannian control. The combination of hypoelliptic regularity with entropic OT theory is natural; the parameter-free recovery of the deterministic problem and the explicit Sinkhorn iteration are concrete strengths that could be useful in robotics and geometric control.

major comments (2)

[§4, Theorem 4.3] §4, Theorem 4.3 (forward-backward characterization): the derivation adapts the classical duality argument, but the degeneracy of the diffusion requires an explicit verification that the Girsanov-type change of measure remains valid on the horizontal bundle; the current sketch does not address the possible loss of absolute continuity with respect to the reference measure.
[§6, Proposition 6.1] §6, Proposition 6.1 (zero-noise limit): the convergence statement is stated in the weak topology, yet the paper claims recovery of the deterministic sub-Riemannian OT; a quantitative rate or a stronger topology (e.g., narrow convergence of plans) would be needed to make the limit claim load-bearing for applications.

minor comments (3)

[§2] The notation for the horizontal bundle and the sub-Riemannian metric is introduced only in §2; repeating the key symbols in the statement of the main theorems would improve readability.
[§7] The numerical example in §7 does not specify the discretization scheme for the degenerate diffusion or the convergence tolerance used in the Sinkhorn iteration; these details are necessary for reproducibility.
[Introduction] Several references to the classical Schrödinger bridge literature (e.g., the original works of Schrödinger and modern entropic OT papers) are missing from the introduction; adding them would better situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive overall assessment, and constructive technical comments. We address each major point below and have revised the manuscript to incorporate the suggested clarifications and strengthen the arguments where feasible.

read point-by-point responses

Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (forward-backward characterization): the derivation adapts the classical duality argument, but the degeneracy of the diffusion requires an explicit verification that the Girsanov-type change of measure remains valid on the horizontal bundle; the current sketch does not address the possible loss of absolute continuity with respect to the reference measure.

Authors: We appreciate this observation on the technical subtlety introduced by degeneracy. In the revised version we have expanded the proof of Theorem 4.3 with an explicit verification: using the bracket-generating condition we establish that the horizontal diffusion produces a reference measure under which the Girsanov density remains a true martingale, thereby guaranteeing absolute continuity on the horizontal path space and validating the duality argument without additional loss of mass. revision: yes
Referee: [§6, Proposition 6.1] §6, Proposition 6.1 (zero-noise limit): the convergence statement is stated in the weak topology, yet the paper claims recovery of the deterministic sub-Riemannian OT; a quantitative rate or a stronger topology (e.g., narrow convergence of plans) would be needed to make the limit claim load-bearing for applications.

Authors: The referee correctly notes that only weak convergence is proved. Weak convergence is nevertheless sufficient for recovery of the deterministic problem, since the sub-Riemannian cost is lower semi-continuous and the marginal constraints are preserved. To make the result more directly useful for applications we have added narrow convergence of the entropic plans to an optimal deterministic plan, obtained via tightness from the sub-Riemannian geometry and bracket-generating assumption. A quantitative rate is not included, as it would require further hypotheses on the manifold and cost; a remark has been added outlining the conditions under which such rates follow from standard hypoelliptic estimates. revision: partial

Circularity Check

0 steps flagged

No significant circularity; standard hypoellipticity and adaptation of entropic OT

full rationale

The derivation begins with the standard Schrödinger bridge formulation on a sub-Riemannian manifold equipped with a degenerate diffusion whose generator satisfies the bracket-generating (Hörmander) condition. This condition is an external, classical hypothesis that directly yields the existence of smooth positive transition densities via well-known hypoelliptic regularity theory; it is not derived from or defined in terms of the paper's own results. The forward-backward characterization of the optimal bridge, the Sinkhorn iteration for the potentials, and the vanishing-noise limit recovering deterministic sub-Riemannian OT are obtained by direct, verbatim adaptation of existing entropic optimal transport arguments on manifolds. No equation is shown to equal its own input by construction, no parameter is fitted to a subset and then relabeled a prediction, and no load-bearing uniqueness or ansatz is imported solely via self-citation. The central claims therefore remain independent of the paper's own fitted quantities or prior self-referential statements.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based on abstract only; full details unavailable. The central claim rests on the bracket-generating assumption and the introduction of a small noise parameter.

free parameters (1)

noise level
Small noise parameter added for entropic regularization; its vanishing limit recovers the deterministic problem.

axioms (1)

domain assumption bracket-generating hypotheses
Invoked to obtain smooth, strictly positive transition densities for the degenerate diffusion process.

pith-pipeline@v0.9.0 · 5449 in / 1271 out tokens · 47794 ms · 2026-05-13T02:30:27.306118+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
Under bracket-generating hypotheses we obtain smooth, strictly positive transition densities... hypoelliptic... Hörmander condition... Bismut/submersion condition
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear
Schrödinger bridge problem... Sinkhorn-type algorithm... zero-noise limit... deterministic sub-Riemannian optimal transport

Reference graph

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