arxiv: 2605.02497 · v1 · submitted 2026-05-04 · 🧮 math.OC

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Closed Forms for Gaussian Kullback--Leibler Unbalanced Optimal Transport without Coupling Entropy

Jiaping Yang , Yunxin Zhang

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Pith reviewed 2026-05-08 17:31 UTC · model grok-4.3

classification 🧮 math.OC

keywords unbalanced optimal transportKullback-Leibler divergenceGaussian measuresRiccati equationquadratic costaffine graph supportclosed form solution

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The pith

Explicit closed-form solutions exist for the Kullback-Leibler unbalanced optimal transport problem between non-degenerate Gaussian measures with quadratic cost and no entropy on the coupling.

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

The paper derives an explicit formula for the static unbalanced optimal transport problem that uses independent Kullback-Leibler penalties to relax the marginal constraints between two finite non-degenerate Gaussian measures. The cost is quadratic, the two relaxation parameters are positive and independent, and the coupling itself carries no entropy penalty. The resulting minimizer takes the form of a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported exactly on an affine graph. A reader would care because this supplies a concrete, computable expression for a problem that would otherwise require numerical optimization, and it directly handles asymmetric relaxations while recovering the ordinary Wasserstein distance in the large-relaxation limit.

Core claim

The minimizer is a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported on an affine graph; the covariance map is the unique positive definite solution of a Riccati equation and admits a principal-square-root representation. The result supplies the modified marginals, the joint minimizer, the transport value, and a direct quadratic KL-dual certificate. In contrast to the entropic Gaussian unbalanced transport case, the optimal plan here is degenerate on the product space. The large-relaxation limit recovers the Gaussian Wasserstein cost for equal masses.

What carries the argument

The covariance map, given as the unique positive definite solution of a Riccati equation that admits a principal square root representation, which determines the affine graph support of the optimal coupling between the adjusted marginals.

If this is right

The optimal plan is supported on an affine graph instead of spreading non-degenerately over the product space as it does in the entropic Gaussian unbalanced transport setting.
The construction treats independent asymmetric Kullback-Leibler relaxations on each marginal, unlike the symmetric equal-penalty Hellinger-Kantorovich case.
Both the adjusted marginals and the exact transport value become available in closed form.
When both relaxation parameters become large, the expression reduces exactly to the quadratic Wasserstein cost between equal-mass Gaussians.

Where Pith is reading between the lines

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

The affine-graph structure may allow direct sampling from the optimal coupling without rejection, which could simplify downstream simulation tasks that rely on Gaussian approximations.
Because the covariance map is obtained from a Riccati equation, standard numerical linear-algebra routines already used in control and filtering can be reused to compute the solution in higher dimensions.
The explicit dual certificate might be inserted into larger optimization problems that nest this unbalanced transport step as a subroutine.

Load-bearing premise

The derivation assumes that a unique positive-definite solution to the Riccati equation for the covariance map exists whenever the input Gaussians are finite and non-degenerate.

What would settle it

For two concrete one-dimensional Gaussians with explicit means and variances, numerically compute the unbalanced transport cost by linear programming on a fine grid and verify whether the value equals the closed-form expression obtained from the Riccati solution and the adjusted marginals.

read the original abstract

We obtain an explicit solution for the static Kullback--Leibler (KL) unbalanced optimal transport problem between finite non-degenerate Gaussian measures with quadratic cost, two independent positive marginal relaxation parameters, and no entropy penalty on the coupling. The minimizer is a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported on an affine graph; in entropic Gaussian unbalanced transport, by contrast, the optimal plan is non-degenerate on the product space. The covariance map is the unique positive definite solution of a Riccati equation and admits a principal-square-root representation. Compared with the known equal-penalty Gaussian Hellinger--Kantorovich endpoint, the result treats the asymmetric two-sided Kullback--Leibler relaxation and gives the modified marginals, joint minimizer, value, and a direct quadratic KL-dual certificate. The large-relaxation limit recovers the Gaussian Wasserstein cost for equal masses.

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

Closed forms for asymmetric KL unbalanced Gaussian OT via Riccati, but the optimality derivation needs explicit verification.

read the letter

The main things to know about this paper are that it derives explicit closed forms for the KL unbalanced optimal transport between Gaussians with asymmetric marginal penalties and no coupling entropy, where the minimizer is a scaled affine-graph Wasserstein coupling and the covariance solves a Riccati equation with square-root representation. It also provides the value and dual certificate, recovering the balanced Wasserstein in the large-relaxation limit. This is new because previous work on Gaussian unbalanced transport either used entropy on the coupling or assumed symmetric penalties. The explicit structure here allows direct computation without solvers for this subclass. The paper does well by giving concrete formulas for the adjusted marginals and the joint plan, plus the consistency check in the limit. This could replace numerical methods in pipelines using Gaussian approximations. The soft spot is the assertion that the Riccati solution is unique and positive definite and satisfies the optimality conditions. The stress-test concern is valid to check: the paper needs to demonstrate that substituting the affine ansatz into the stationarity conditions yields a well-posed Riccati that is globally minimizing for all positive asymmetric parameters, without implicit restrictions. If the derivations in the full text do this rigorously, the result is solid; if not, there might be gaps. This paper is for people working on analytical solutions in optimal transport, particularly unbalanced variants with Gaussians. Readers who need fast, exact expressions for such problems will get value from it. It shows clear engagement with the literature on Gaussian OT and unbalanced transport. I recommend sending it to peer review so the derivations can be verified by experts.

Referee Report

2 major / 2 minor

Summary. The manuscript derives an explicit closed-form solution for the static Kullback-Leibler unbalanced optimal transport problem between finite non-degenerate Gaussian measures with quadratic cost, independent positive marginal relaxation parameters, and no coupling entropy. The minimizer is a scaled Wasserstein coupling between adjusted Gaussian marginals supported on an affine graph; the covariance map is the unique positive definite solution of a Riccati equation admitting a principal square-root representation. The result supplies modified marginals, the joint minimizer, the transport value, and a quadratic KL-dual certificate, recovering the balanced Gaussian Wasserstein cost in the large-relaxation limit and contrasting with non-degenerate plans in the entropic setting.

Significance. If the central derivation holds, the result supplies a concrete computational and theoretical advance for asymmetric unbalanced OT with Gaussians, extending the equal-penalty Hellinger-Kantorovich endpoint and the balanced Wasserstein case. The explicit Riccati structure and affine-graph support enable direct evaluation without numerical optimization and highlight geometric differences from entropic formulations. The provision of the dual certificate and limit recovery are additional strengths that make the closed form falsifiable and immediately usable in applications.

major comments (2)

[Derivation of the covariance map (likely §3–4)] The central claim that the covariance map is the unique positive definite solution of a Riccati equation obtained from the stationarity conditions of the primal unbalanced KL objective (quadratic cost plus two independent marginal KL penalties) requires an explicit verification step. The manuscript must show that substituting the affine-graph ansatz into the Euler-Lagrange equations or dual certificate produces the Riccati equation, that the resulting algebraic equation remains uniquely solvable for all positive asymmetric relaxation parameters, and that the solution globally minimizes the objective (see the skeptic note on first-order optimality conditions).
[Assumptions and well-posedness (likely §2)] The necessity of the non-degenerate finite Gaussian assumption for both the affine-graph support and the existence of the principal-square-root representation is asserted but not independently verified against the first-order conditions; the manuscript should include a short argument confirming that degeneracy or infinite-mass cases break the closed form in a manner consistent with the stated hypotheses.

minor comments (2)

Notation for the two independent relaxation parameters should be introduced once with a single consistent symbol pair and reused verbatim in all subsequent equations and statements.
The comparison paragraph with the equal-penalty Hellinger-Kantorovich case would benefit from an explicit side-by-side table of the resulting marginal adjustments and covariance maps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address each major comment below and will revise the paper to incorporate the requested clarifications.

read point-by-point responses

Referee: [Derivation of the covariance map (likely §3–4)] The central claim that the covariance map is the unique positive definite solution of a Riccati equation obtained from the stationarity conditions of the primal unbalanced KL objective (quadratic cost plus two independent marginal KL penalties) requires an explicit verification step. The manuscript must show that substituting the affine-graph ansatz into the Euler-Lagrange equations or dual certificate produces the Riccati equation, that the resulting algebraic equation remains uniquely solvable for all positive asymmetric relaxation parameters, and that the solution globally minimizes the objective (see the skeptic note on first-order optimality conditions).

Authors: We agree that an explicit verification step will strengthen the presentation. In the revised manuscript we will insert a dedicated paragraph (or short subsection) in §3 that substitutes the affine-graph ansatz directly into the first-order stationarity conditions obtained from the primal objective, derives the Riccati equation, proves uniqueness of the positive-definite solution for arbitrary positive asymmetric relaxation parameters by appealing to the monotonicity properties of the Riccati map and the principal square-root representation, and confirms global minimality by strict convexity of the unbalanced KL functional together with uniqueness of the critical point. revision: yes
Referee: [Assumptions and well-posedness (likely §2)] The necessity of the non-degenerate finite Gaussian assumption for both the affine-graph support and the existence of the principal-square-root representation is asserted but not independently verified against the first-order conditions; the manuscript should include a short argument confirming that degeneracy or infinite-mass cases break the closed form in a manner consistent with the stated hypotheses.

Authors: We will add a concise paragraph in §2 that verifies necessity against the first-order conditions. We will show that singularity of either covariance matrix violates the affine-graph support condition required by the stationarity equations and renders the principal square root non-unique or undefined in the required positive-definite sense; likewise, infinite-mass limits make at least one KL penalty infinite, causing the objective to diverge and the closed-form expression to cease to apply. These limiting arguments will be tied directly to the Euler-Lagrange system to confirm consistency with the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from KL objective to independent Riccati solution

full rationale

The paper derives a closed-form solution for the unbalanced KL optimal transport between Gaussians by substituting an affine-graph ansatz into the stationarity conditions of the primal problem (quadratic cost plus independent marginal KL penalties), yielding a Riccati equation whose positive-definite solution is then represented via principal square root. No step reduces by construction to a fitted parameter, renamed input, or self-citation chain; the uniqueness claim is presented as following from the algebraic properties of the derived Riccati under the stated non-degeneracy assumptions. The abstract and description contain no self-referential fitting or load-bearing citations to prior author work that would collapse the central result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions for Gaussian measures and quadratic-cost optimal transport; no new entities are introduced and no free parameters are fitted beyond the given relaxation parameters.

axioms (2)

domain assumption Input measures are finite non-degenerate Gaussian measures.
Explicitly required for the problem statement and for the existence of the positive-definite Riccati solution.
domain assumption Quadratic cost is used.
Standard choice in the Gaussian OT literature but necessary for the Riccati structure to appear.

pith-pipeline@v0.9.0 · 5458 in / 1547 out tokens · 47962 ms · 2026-05-08T17:31:28.160867+00:00 · methodology

discussion (0)

Reference graph

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