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arxiv: 2605.23608 · v1 · pith:YIAXDUSEnew · submitted 2026-05-22 · 🧮 math.OC · math.DG

Linear quadratic optimal transport and interpolation inequalities

Luca Rizzi , Alec Jacopo Almo Schiavoni Piazza This is my paper

Pith reviewed 2026-05-25 04:09 UTC · model grok-4.3

classification 🧮 math.OC math.DG
keywords optimal transportMonge problemlinear quadratic costsdisplacement interpolationentropy inequalitiessub-Riemannian geometry
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The pith

Linear-quadratic costs make the Monge problem well-posed with regular optimal maps even for non-negative costs and produce entropy interpolation inequalities for displacement interpolations.

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

The paper establishes that the Monge problem is well-posed for linear-quadratic costs in optimal control systems, with the optimal transport map being regular. This extends the 2011 results to the case where the cost is non-negative. It also derives general interpolation inequalities for entropy functionals from the displacement interpolation of measures. These results are motivated by the role of linear-quadratic systems as model spaces in sub-Riemannian geometry comparison theory.

Core claim

For linear-quadratic costs the Monge problem is well-posed and the optimal transport map is regular, extending the results of Hindawi, Pomet and Rifford 2011 to non-negative costs. Displacement interpolations satisfy general entropy interpolation inequalities. The analysis is motivated by the role of these systems as natural model spaces for comparison theory in sub-Riemannian geometry.

What carries the argument

The linear-quadratic structure of costs in optimal control systems, supplying the convexity and regularity for well-posedness and interpolation inequalities.

If this is right

  • The Monge problem admits solutions given by regular maps for non-negative linear-quadratic costs.
  • Displacement interpolations obey general entropy interpolation inequalities.
  • These properties extend prior results on strictly positive costs.

Where Pith is reading between the lines

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

  • The inequalities may enable new comparison theorems in sub-Riemannian geometry.
  • The approach could be adapted to other optimal control systems with similar convexity properties.
  • Explicit calculations for specific measures like Gaussians would provide concrete instances of the inequalities.

Load-bearing premise

The linear-quadratic structure supplies enough convexity and regularity to guarantee existence and uniqueness of the optimal map even when the cost is non-negative.

What would settle it

Constructing a non-negative linear-quadratic cost for which there is no unique regular optimal transport map would disprove the extension of the well-posedness result.

read the original abstract

This paper investigates the optimal transport problem within the framework of Linear Quadratic optimal control systems. We establish the well-posedness of the Monge problem and analyze the regularity of the resulting optimal transport map, extending the results obtained in [Hindawi, Pomet, Rifford, 2011] for non-negative costs. Furthermore, we study the displacement interpolation of measures and derive general interpolation inequalities for entropy functionals. Our analysis is motivated by the role of these systems as natural model spaces for comparison theory in sub-Riemannian geometry.

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

1 major / 2 minor

Summary. The paper claims to establish well-posedness of the Monge problem and regularity of the optimal transport map for linear-quadratic costs that are non-negative (possibly degenerate), extending the 2011 results of Hindawi, Pomet, and Rifford. It further studies displacement interpolations of measures and derives general interpolation inequalities for entropy functionals, motivated by applications to comparison theory in sub-Riemannian geometry.

Significance. If the central claims hold, this work would provide a valuable extension of optimal transport theory to possibly degenerate linear-quadratic settings, serving as model spaces for sub-Riemannian comparison theorems and yielding new entropy interpolation inequalities. The extension to non-negative costs is particularly relevant for geometric applications where strict positivity may not hold.

major comments (1)
  1. [Main well-posedness theorem (likely §3 or §4)] Main theorem on well-posedness of the Monge problem (extension of Hindawi-Pomet-Rifford 2011): The argument for uniqueness of the optimal map when the quadratic form is only positive semi-definite must be made explicit. The 2011 result relies on strict convexity to guarantee uniqueness via standard OT arguments; when the cost vanishes along nontrivial directions the functional may fail to be strictly convex on the space of measures, and the manuscript needs to supply either a direct proof that uniqueness persists or an additional selection principle. This is load-bearing for the claimed extension to non-negative costs.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should clarify whether the non-negative costs are assumed to satisfy any additional structural conditions (e.g., controllability of the underlying linear system) that restore uniqueness.
  2. [§2 (Preliminaries)] Notation for the linear-quadratic cost functional and the associated quadratic form should be introduced with explicit matrix assumptions early in the preliminaries.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to clarify the uniqueness argument in the main well-posedness theorem. We address this point below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: Main theorem on well-posedness of the Monge problem (extension of Hindawi-Pomet-Rifford 2011): The argument for uniqueness of the optimal map when the quadratic form is only positive semi-definite must be made explicit. The 2011 result relies on strict convexity to guarantee uniqueness via standard OT arguments; when the cost vanishes along nontrivial directions the functional may fail to be strictly convex on the space of measures, and the manuscript needs to supply either a direct proof that uniqueness persists or an additional selection principle. This is load-bearing for the claimed extension to non-negative costs.

    Authors: We agree that the uniqueness statement for positive semi-definite quadratic forms requires an explicit argument rather than an implicit appeal to the 2011 strict-convexity case. Our proof proceeds by reducing to the controllable subspace orthogonal to the kernel of the quadratic form, then invoking the linear structure of the control system to obtain a measurable selection that yields a unique optimal map. To make this fully transparent, we will add a dedicated lemma (with a self-contained proof) immediately preceding the main theorem that isolates the degeneracy handling and shows that the functional remains strictly convex on the relevant quotient space. This change will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: direct extension of independent external theorem

full rationale

The manuscript extends well-posedness and regularity results from the 2011 Hindawi-Pomet-Rifford paper (distinct authors) to the non-negative cost case. No self-citations, fitted parameters, ansatzes, or self-definitional steps appear in the abstract or described derivation. The central claims rest on external prior theorems rather than reducing to the paper's own inputs by construction. This is the normal case of an honest extension.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard well-posedness theory of linear-quadratic optimal control and on the displacement-interpolation framework of optimal transport; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Linear-quadratic optimal control systems admit sufficiently regular value functions and optimal trajectories
    Invoked to guarantee existence and regularity of the transport map.

pith-pipeline@v0.9.0 · 5607 in / 1122 out tokens · 23638 ms · 2026-05-25T04:09:35.895844+00:00 · methodology

discussion (0)

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