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arxiv: 2605.11429 · v2 · pith:22ZGY44Nnew · submitted 2026-05-12 · 🧮 math.OC

From Schrodinger Bridge to Optimal Transport over Sub-Riemannian Manifolds

Daniel Owusu Adu , Karthik Elamvazhuthi , Bahman Gharesifard This is my paper

Pith reviewed 2026-05-19 18:07 UTC · model grok-4.3

classification 🧮 math.OC
keywords Schrödinger bridgesub-Riemannian manifoldsoptimal transportentropic regularizationdegenerate diffusionSinkhorn algorithmdistribution steeringbracket-generating
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The pith

Entropic regularization by control-aligned noise turns sub-Riemannian optimal transport into a tractable Schrödinger bridge problem.

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

The paper studies the least-energy reshaping of probability distributions under sub-Riemannian constraints by introducing small noise aligned with the control directions. This converts the problem into a Schrödinger bridge on a degenerate diffusion whose transition densities are smooth and positive when the manifold is bracket-generating. The bridge admits a forward-backward characterization that enables a Sinkhorn-type algorithm for the potentials. As the noise vanishes the formulation recovers the original deterministic optimal transport problem, providing a practical numerical method illustrated by an example.

Core claim

Under bracket-generating hypotheses the optimal bridge for the degenerate diffusion admits a forward-backward characterization, yielding a Sinkhorn algorithm for the Schrödinger potentials; in the zero-noise limit this recovers the deterministic sub-Riemannian optimal transport between the given marginals.

What carries the argument

The Schrödinger bridge problem associated to the degenerate diffusion obtained by adding noise along the horizontal directions, which regularizes the sub-Riemannian optimal transport.

If this is right

  • The transition densities of the reference process are smooth and strictly positive.
  • A practical Sinkhorn-type algorithm computes the optimal Schrödinger potentials.
  • The deterministic sub-Riemannian optimal transport is recovered as the noise level vanishes.
  • This provides a numerically tractable formulation for density control over underactuated systems.

Where Pith is reading between the lines

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

  • The method could be tested on other bracket-generating structures such as the Heisenberg group to verify convergence rates.
  • Similar regularization might apply to optimal transport on more general manifolds with degenerate metrics.
  • Extensions to time-dependent or nonlinear control systems could follow from the same forward-backward structure.

Load-bearing premise

The manifold satisfies bracket-generating hypotheses so that the added noise produces a diffusion with smooth strictly positive transition densities.

What would settle it

A concrete falsifier would be finding a bracket-generating sub-Riemannian manifold where the transition density of the degenerate diffusion is not strictly positive at some positive time.

Figures

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

Figure 1
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. 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. 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. 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.

Referee Report

2 major / 2 minor

Summary. The paper studies entropic regularization of sub-Riemannian optimal transport by adding small noise aligned with horizontal control directions, formulating the problem as a Schrödinger bridge on the manifold. Under bracket-generating hypotheses it claims to obtain smooth strictly positive transition densities for the resulting degenerate diffusion, a forward-backward characterization of the optimal bridge, a practical Sinkhorn-type algorithm for the Schrödinger potentials, and recovery of the deterministic sub-Riemannian OT problem in the zero-noise limit, illustrated by a numerical example.

Significance. If the central claims are rigorously substantiated, the work would connect Schrödinger-bridge techniques to sub-Riemannian geometry and provide a numerically tractable method for distribution steering under nonholonomic constraints, with potential impact on geometric control and entropic optimal transport.

major comments (2)
  1. [Abstract and main results] Abstract and main results section: the load-bearing claim that bracket-generating hypotheses yield smooth, strictly positive transition densities p_t(x,y)>0 for all x,y and t>0 (needed to justify the forward-backward characterization and Sinkhorn algorithm) is invoked but not accompanied by an explicit support-theorem argument or verification for the specific horizontal-noise degenerate diffusion; hypoellipticity gives smoothness but strict positivity requires separate controllability/support analysis.
  2. [Vanishing-noise limit] Section on the vanishing-noise limit: the recovery of the deterministic sub-Riemannian OT problem is asserted without derivation steps, error bounds, or explicit verification that the entropic minimizer converges to the sub-Riemannian geodesic problem; the abstract states the claim but supplies no equations or analysis showing the reduction.
minor comments (2)
  1. [Setup] Clarify the precise form of the degenerate diffusion generator and the horizontal noise alignment in the setup section to aid readability.
  2. [Introduction] Add a short discussion of related literature on hypoelliptic Schrödinger bridges or support theorems if not already present.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and main results] Abstract and main results section: the load-bearing claim that bracket-generating hypotheses yield smooth, strictly positive transition densities p_t(x,y)>0 for all x,y and t>0 (needed to justify the forward-backward characterization and Sinkhorn algorithm) is invoked but not accompanied by an explicit support-theorem argument or verification for the specific horizontal-noise degenerate diffusion; hypoellipticity gives smoothness but strict positivity requires separate controllability/support analysis.

    Authors: We agree that hypoellipticity alone (from Hörmander's theorem under bracket-generating conditions) ensures smoothness of the transition densities but does not automatically guarantee strict positivity. The latter follows from controllability of the horizontal vector fields, which implies that the support of the degenerate diffusion is the full manifold for any t>0. In the revised manuscript we will add an explicit paragraph in the section introducing the reference process, invoking the support theorem for hypoelliptic diffusions on manifolds and citing the relevant controllability results from sub-Riemannian geometry. This will make the justification for the forward-backward characterization and Sinkhorn algorithm fully explicit. revision: yes

  2. Referee: [Vanishing-noise limit] Section on the vanishing-noise limit: the recovery of the deterministic sub-Riemannian OT problem is asserted without derivation steps, error bounds, or explicit verification that the entropic minimizer converges to the sub-Riemannian geodesic problem; the abstract states the claim but supplies no equations or analysis showing the reduction.

    Authors: We acknowledge that the vanishing-noise limit is stated concisely and would benefit from additional detail. The claim relies on the standard Gamma-convergence of the entropic cost to the unregularized sub-Riemannian transport cost as the noise parameter tends to zero. In the revision we will expand the relevant section with a sketch of the argument, including the key limiting equations and a statement on convergence of the optimal bridges to sub-Riemannian geodesics. Full quantitative error bounds are technically involved and will be noted as a direction for future work rather than fully derived here. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation rests on external hypoellipticity and Schrödinger bridge results

full rationale

The paper invokes bracket-generating hypotheses to obtain smooth strictly positive transition densities for the degenerate diffusion, then applies the standard forward-backward Schrödinger bridge characterization to derive a Sinkhorn algorithm and the zero-noise limit recovering deterministic sub-Riemannian OT. These steps cite established results from sub-Riemannian geometry (Hörmander hypoellipticity, Chow controllability for support) and classical Schrödinger bridge theory rather than defining the target quantities in terms of themselves or fitting parameters to the same data. No equations reduce the claimed recovery or algorithm to a self-referential fit, and no load-bearing self-citation chain is present. The construction is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard sub-Riemannian geometry assumptions plus the modeling choice of noise alignment; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The sub-Riemannian manifold is bracket-generating
    Invoked to guarantee existence of smooth strictly positive transition densities for the degenerate diffusion.

pith-pipeline@v0.9.0 · 5680 in / 1115 out tokens · 33980 ms · 2026-05-19T18:07:52.138064+00:00 · methodology

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Lean theorems connected to this paper

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  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
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    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Under bracket-generating hypotheses we obtain smooth, strictly positive transition densities and a forward–backward characterization of the optimal bridge... the reference process is a degenerate diffusion on the sub-Riemannian manifold.

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