Optimal Transport on Lie Group Orbits

Bahar Taskesen

arxiv: 2508.09377 · v2 · pith:K2EQJFX7new · submitted 2025-08-12 · 🧮 math.OC · math.GR· math.PR

Optimal Transport on Lie Group Orbits

Bahar Taskesen This is my paper

Pith reviewed 2026-05-22 13:14 UTC · model grok-4.3

classification 🧮 math.OC math.GRmath.PR

keywords optimal transportLie group orbitsstabilizer subgroupMonge problemc-convexityCartan decompositionWishart distributionsquadratic cost

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The pith

Optimal transport between measures on a Lie group orbit reduces to optimization over the stabilizer subgroup, with the bound tight when a c-convex certificate holds.

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

The paper identifies symmetry as the source of closed-form solutions in optimal transport by modeling it through Lie group actions on the outcome space. When two measures lie on the same orbit of a reference distribution, the Monge problem reduces to a lower-dimensional optimization over the stabilizer subgroup, yielding an explicit upper bound on the transport cost. Under mild regularity conditions, a feasible point from this reduced problem whose induced map satisfies a c-convex first-order certificate makes the bound tight for both Monge and Kantorovich formulations, with the optimal map realized by a group element. For quadratic cost and affine actions, the certificate simplifies to an algebraic self-adjoint positive-semidefinite condition on the linear part, which Cartan theory can guarantee under a global decomposition and fixed-point subgroup containment. This unifies known solutions for elliptical distributions and produces new closed forms for Wishart, inverse-Wishart, and matrix beta type II distributions under squared Frobenius cost.

Core claim

Fixing a Lie group action and reference distribution, the Monge problem for measures on the same orbit admits an explicit upper bound from optimization over the stabilizer subgroup. A feasible point whose induced transport map satisfies the c-convex first-order certificate makes this upper bound tight for both Monge and Kantorovich problems, realizing the optimal map as a group element. For quadratic cost on finite-dimensional Hilbert space with affine-induced actions, the certificate reduces to the candidate map having self-adjoint positive semidefinite linear part, guaranteed when the linear image admits a global Cartan decomposition and the fixed-point subgroup is contained in the linear

What carries the argument

The reduced optimization problem over the stabilizer subgroup of the reference distribution, which generates candidate transport maps realized by group elements and provides a tight upper bound under the c-convex first-order certificate.

If this is right

The Monge problem admits an explicit upper bound given by optimization over the stabilizer subgroup.
When the c-convex certificate holds, the upper bound is tight for both Monge and Kantorovich formulations with the optimal map a group element.
For quadratic cost and affine actions, the condition reduces to the linear part being self-adjoint and positive semidefinite.
A structural criterion from Cartan theory guarantees the algebraic condition when the group admits a global decomposition and fixed-point containment holds.
New closed-form solutions exist for Wishart, inverse-Wishart, and matrix beta type II distributions under squared Frobenius cost.

Where Pith is reading between the lines

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

The same orbit-reduction approach may extend to derive explicit maps for other matrix-variate or manifold-valued distributions invariant under specific Lie group actions.
This stabilizer viewpoint could connect to symmetry reductions in related infinite-dimensional problems such as Wasserstein barycenters or gradient flows on orbits.
For concrete groups like the orthogonal group, the reduced problem might admit efficient numerical solvers that scale with stabilizer dimension rather than ambient space.

Load-bearing premise

The candidate map obtained from the stabilizer optimization must satisfy the c-convex first-order certificate (or the algebraic self-adjoint positive-semidefinite condition for quadratic cost).

What would settle it

A concrete pair of measures on the same orbit where the stabilizer optimization yields a feasible point, the induced map fails the c-convex certificate, and the true optimal transport cost is strictly lower than the computed upper bound.

read the original abstract

In its most general form, the optimal transport problem is an infinite-dimensional optimization problem, yet certain notable instances admit closed-form solutions. We identify the common source of this tractability as \textit{symmetry} and formalize it using Lie group theory. Fixing a Lie group action on the outcome space and a reference distribution, we study optimal transport between measures lying on the same Lie group orbit of the reference distribution. In this setting, the Monge problem admits an explicit upper bound given by an optimization problem over the stabilizer subgroup of the reference distribution. The reduced problem's dimension scales with that of the stabilizing subgroup and, in the tractable cases we study, is either zero or finite. Under mild regularity conditions, a feasible point of this reduced problem whose induced transport map satisfies a $c$-convex first-order certificate makes the upper bound tight for both the Monge and Kantorovich formulations, with the optimal map realized by a group element. For the quadratic cost on a finite-dimensional Hilbert space and affine-induced actions, the $c$-convex certificate reduces to an algebraic condition: the candidate map must have self-adjoint positive semidefinite linear part. We give a structural criterion, based on Cartan theory, that guarantees this condition. When the linear image of the acting group admits a global Cartan decomposition and its fixed-point subgroup is contained in the linear image of the stabilizer of the reference law, the compact component can be absorbed by the stabilizer, yielding a transport map with a self-adjoint positive definite linear part. This orbit-based viewpoint unifies known closed-form solutions, such as elliptical distributions, and yields new closed-form solutions for Wishart, inverse-Wishart, and matrix beta type~II distributions under the squared Frobenius cost.

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

Lie group orbit reduction yields new closed-form OT maps for Wishart laws but tightness stays conditional on the certificate holding.

read the letter

The main takeaway is that measures on the same Lie group orbit let you reduce the Monge problem to an optimization over the stabilizer subgroup, and a point whose induced map meets the c-convex certificate makes the bound tight for both Monge and Kantorovich versions. For quadratic cost the certificate simplifies to a self-adjoint positive-semidefinite linear part, and Cartan theory supplies a structural guarantee when the linear image has a global decomposition and the fixed-point subgroup sits inside the stabilizer image. This produces explicit transport maps for Wishart, inverse-Wishart, and matrix-beta-II distributions under squared Frobenius cost that were not in the prior literature, while also recovering the elliptical cases in one framework. The reduction is clean, the dimension drop is practical, and the construction stays grounded in standard group actions without fitted parameters or circular steps. The central claim is explicitly conditional on the existence of a certificate-satisfying feasible point, which the Cartan criterion addresses under the stated inclusions. A modest soft spot is that the abstract sketches the application to the new distributions but leaves the explicit verification of the fixed-point-subgroup inclusion and the resulting maps somewhat compressed; seeing the full derivation would remove any remaining doubt about applicability. No load-bearing gaps or internal contradictions appear in the logic. This is for researchers in optimal transport or multivariate statistics who need closed forms for symmetric or matrix-valued laws. It shows clear engagement with the literature and concrete additions, so it deserves a serious referee rather than a desk reject. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The paper introduces a Lie-group-theoretic framework for optimal transport between measures lying on the same orbit of a reference distribution under a group action. It reduces the Monge problem to a finite- or zero-dimensional optimization over the stabilizer subgroup of the reference measure. Under mild regularity conditions, a feasible point whose induced map satisfies a c-convex first-order certificate renders the upper bound tight for both Monge and Kantorovich problems, with the optimal map realized by a group element. For quadratic cost on Hilbert space with affine actions, the certificate reduces to an algebraic self-adjoint positive-semidefinite condition on the linear part; a Cartan-theoretic structural criterion is given that guarantees this when the linear image admits a global Cartan decomposition and the fixed-point subgroup is contained in the stabilizer image. The framework unifies known closed-form solutions (e.g., elliptical distributions) and yields new explicit solutions for Wishart, inverse-Wishart, and matrix-beta type II laws under squared Frobenius cost.

Significance. If the central claims hold, the work supplies a systematic symmetry-based method for identifying and solving tractable OT instances, explaining the origin of closed-form solutions in the literature and generating new ones for matrix-variate distributions that arise in statistics. The explicit reduction to stabilizer optimization and the Cartan-derived algebraic certificate constitute a genuine contribution that bridges Lie theory with OT duality; the parameter-free character of the construction and the conditional but falsifiable nature of the tightness criterion are particular strengths.

major comments (1)

[Section presenting the Cartan-theoretic structural criterion (quadratic-cost case)] Section presenting the Cartan-theoretic structural criterion (quadratic-cost case): the claim that the compact component can be absorbed by the stabilizer when the linear image admits a global Cartan decomposition and the fixed-point subgroup is contained in the linear image of the stabilizer is load-bearing for the new closed-form results; the manuscript must supply an explicit verification of this inclusion for the Wishart, inverse-Wishart, and matrix-beta type II distributions rather than relying on an implicit appeal to standard facts.

minor comments (2)

[Introduction and notation] The notation distinguishing the acting group, its linear image, the stabilizer, and the induced transport map should be introduced with a low-dimensional running example (e.g., orthogonal group acting on covariance matrices) to improve accessibility before the general theorems.
[Main reduction theorem] A short table or diagram summarizing the reduction from the original OT problem to the stabilizer problem, together with the precise statement of the c-convex certificate, would help readers track the logical flow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the contribution, and constructive suggestion regarding the Cartan-theoretic criterion. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: Section presenting the Cartan-theoretic structural criterion (quadratic-cost case): the claim that the compact component can be absorbed by the stabilizer when the linear image admits a global Cartan decomposition and the fixed-point subgroup is contained in the linear image of the stabilizer is load-bearing for the new closed-form results; the manuscript must supply an explicit verification of this inclusion for the Wishart, inverse-Wishart, and matrix-beta type II distributions rather than relying on an implicit appeal to standard facts.

Authors: We agree that the inclusion is load-bearing for the new closed-form results and that an explicit verification strengthens the manuscript. In the revised version we will add a short dedicated paragraph (or appendix entry) for each distribution. For the Wishart law we will compute the fixed-point subgroup of the linear action explicitly and verify its containment in the image of the stabilizer by direct matrix calculation. The same direct verification will be supplied for the inverse-Wishart and matrix-beta type II cases, confirming that the fixed-point subgroup lies inside the linear image of the stabilizer without relying on implicit appeals. These additions will be placed immediately after the statement of the structural criterion so that the application to the three families is fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard Lie theory

full rationale

The paper constructs an upper bound for the Monge problem directly from the Lie group action on the outcome space and a fixed reference distribution, reducing it to optimization over the stabilizer subgroup whose dimension is determined by the group structure. Tightness is established conditionally on the existence of a feasible point whose induced map satisfies an independently verifiable c-convex first-order certificate (or the algebraic self-adjoint PSD condition for quadratic cost), with the latter guaranteed by a structural criterion drawn from standard Cartan theory rather than any internal fit or self-referential definition. No step renames a fitted quantity as a prediction, imports uniqueness solely via author self-citation, or smuggles an ansatz; the unification of elliptical distributions and derivation of new closed forms for Wishart-type laws follow from applying the orbit framework to specific measures without reducing the target result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard Lie-group and Cartan theory plus the existence of a c-convex certificate; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption The outcome space carries a Lie group action that preserves the reference distribution.
Invoked in the opening setup of the orbit-based OT problem.
domain assumption Mild regularity conditions hold so that the c-convex first-order certificate is well-defined.
Stated explicitly as the condition under which the upper bound becomes tight.

pith-pipeline@v0.9.0 · 5845 in / 1527 out tokens · 30947 ms · 2026-05-22T13:14:26.910214+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Under mild regularity conditions, a feasible point of the reduced stabilizer problem whose induced transport map satisfies a c-convex first-order certificate makes the upper bound tight... (Theorem 4.1, §4)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We identify the common source of this tractability as symmetry and formalize it using Lie group theory... orbit G#ρ and stabilizer StabG(ρ)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.