How sensitively does an optimal transport map depend on its target?

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Family 374 · Partial differential equations
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

An optimal transport map rearranges one probability distribution into another with the least mean squared travel distance. If the target distribution changes only a little, should the optimal map change only a little as well?

Continuity alone gives no rate. A numerical or statistical application needs to know how an error in the target distribution translates into an error in the map assigning destinations to individual source points.

Family 374 claims the sharp uniform rate when the source is uniformly distributed on a compact convex body. In every dimension at least two, the map varies Hölder continuously with exponent one-third in the target’s quadratic Wasserstein distance. The constant works for every target in a fixed compact set, including discrete targets with vanishingly small atom masses.

A three-atom construction on a fixed cube is claimed to show that no larger exponent works uniformly. In particular, it disproves a conjectured square-root rate in this class.

2. Background and essential definitions

The Brenier map

Let $K\subset\mathbb R^d$ be compact and convex with nonempty interior. The fixed source probability measure is

$$\rho=\frac{\mathbf1_K}{|K|}\,dx,$$

where $|K|$ is its volume. Let $\mu$ be a target probability measure supported in a compact set $Y$.

Among maps $T$ satisfying $T_\#\rho=\mu$, quadratic optimal transport minimizes

$$\int_K|x-T(x)|^2\,d\rho(x).$$

The pushforward condition means that if $x$ is sampled from $\rho$, then $T(x)$ has distribution $\mu$. Brenier’s theorem gives an almost-everywhere unique optimal map of the form

$$T_\mu=\nabla\varphi_\mu,$$

where $\varphi_\mu$ is convex. The potential is determined only up to an additive constant; its gradient is unaffected by that freedom.

Even though the source has a density, the target may consist of atoms. In that case the map is typically constant on cells and jumps across their boundaries. Stability here cannot rely on every transport map being a smooth function.

Two different distances

The quadratic Wasserstein distance between targets is

$$W_2(\mu,\nu)= \inf_{\pi\in\Pi(\mu,\nu)} \left(\int|y-z|^2\,d\pi(y,z)\right)^{1/2}.$$

A coupling $\pi$ matches target mass from $\mu$ to target mass from $\nu$. It may choose whichever matching is cheapest.

The map error instead compares the destinations of the same source point:

$$\|T_\mu-T_\nu\|_{L^2(\rho)} =\left(\int_K|T_\mu(x)-T_\nu(x)|^2\,d\rho(x)\right)^{1/2}.$$

These are different comparisons. Two target distributions can be close because their atoms barely move, while their optimal source partitions change enough to send some source points to entirely different atoms.

A Hölder bound with exponent $\alpha$ has the form

$$\|T_\mu-T_\nu\|_{L^2(\rho)}\le C W_2(\mu,\nu)^\alpha.$$

For small distances, a larger exponent gives a stronger bound. “Uniform” means one constant works for every pair of targets in the prescribed class.

3. Development history and previous partial results

Brenier’s polar-factorization theorem supplies the representation by convex gradients. Quantitative stability asks how reliably that representation changes when the measure changes.

Early estimates included a local square-root bound along target curves near regular data, attributed to Ambrosio and recorded by Gigli. Berman obtained quantitative estimates under geometric and density assumptions in a different source–target variation setting.

Mérigot, Delalande and Chazal used duality to obtain target-uniform bounds, including a dimension-independent exponent $2/15$ measured in $W_1$. Here $W_1$ is the optimal expected travel distance without squaring it.

Delalande and Mérigot obtained an $L^2$ map bound of order $W_1^{1/6}$ for source densities bounded above and below on convex domains. Their route—first compare potentials, then interpolate to control gradients—is an important predecessor of this manuscript’s final step. Letrouit and Mérigot later extended potential stability to John domains using local-to-global methods.

Divol, Niles-Weed and Pooladian obtained a one-third $W_2$ bound for regular sources and nondegenerate finite targets. Its constant depends on atom count, lower mass bounds and geometric separation conditions. Thus the exponent itself has a precedent; the present theorem claims a constant uniform even as those target features degenerate.

At the regular end of the subject, Caja-Lopez, Delgadino and Kitagawa obtain Lipschitz stability under strong positive-density and domain-regularity hypotheses. Better rates in such classes are compatible with a worse optimal rate when arbitrary atomic targets are allowed.

For unrestricted targets, recent work also gives stronger $W_1$ estimates, including a sharp quarter-power bound of Mérigot reported in the literature cited by the manuscript. Since $W_1\le W_2$, that comparison alone does not give a one-third $W_2$ estimate.

Letrouit exhibited a one-third obstruction for a nonconvex source and conjectured a uniform square-root rate for a uniform convex source. Family 374 claims that convexity does not remove the obstruction: three atoms on a fixed cube already force one-third, and its upper bound attains that endpoint.

4. What the new manuscript claims

For every compact convex body $K\subset\mathbb R^d$ with interior, every nonempty compact set $Y\subset\mathbb R^d$, and every $d\ge2$, there is a finite constant $C(K,Y)$ such that

$$\|T_\mu-T_\nu\|_{L^2(\rho)} \le C(K,Y)W_2(\mu,\nu)^{1/3}$$

for all Borel probability measures $\mu,\nu$ supported in $Y$.

Targets may be atomic, singular or absolutely continuous. The constant is independent of the number of atoms, their positive masses, site separation and target density bounds. It can depend on the fixed source and target sets, hence on their geometry and dimension.

No smoothness or strict convexity of $K$ is required. The compact set $Y$ need not be convex. The source, however, is specifically uniform on $K$; the theorem does not assert this same endpoint for every nonuniform bounded source density.

Sharpness is stated on $K=Y=[-1,1]^d$. Even restricting each target to exactly three atoms, every exponent larger than $1/3$ fails with a target-independent constant.

5. Why the advance matters

Importance: 4/5. If correct, the result determines the optimal uniform stability exponent for a broad and useful class of quadratic transport maps, while resolving the proposed square-root improvement negatively.

The uniformity is practically significant. Discrete approximations can introduce tiny masses, nearly merging sites or changing atom counts. A bound whose constant deteriorates with those features may be unsuitable for a refinement limit. The claimed estimate remains valid throughout those degenerations.

The theorem also quantifies the limits of representing a distribution by its optimal map from one fixed reference measure. That representation is stable, but its worst-case sensitivity is inherently nonlinear. Additional target regularity can improve the rate; arbitrary compact support alone cannot.

6. The main idea

A thin cell creates the sharp obstruction

On the uniform cube, choose $0<a<1/2$, let $b=a/2$, and consider convex potentials

$$\varphi_a(x)=\max\{-x_1,x_1,a\},$$

and

$$\widetilde\varphi_a(x)=\max\{-x_1,x_1,a+bx_2\}.$$

A maximum of affine functions is convex. Its gradient equals the slope of the winning affine function almost everywhere, so each potential defines a three-valued optimal map to its own target law.

For the first potential, the central cell is $|x_1|<a$. Tilting the central affine function changes that cell to $|x_1|<a+bx_2$. Its total mass remains $a$, but some source points switch between the central and outer destinations.

The targets are

$$\mu_a=\frac{1-a}{2}(\delta_{-e_1}+\delta_{e_1})+a\delta_0,$$

$$\nu_a=\frac{1-a}{2}(\delta_{-e_1}+\delta_{e_1})+a\delta_{be_2}.$$

Here $\delta_y$ denotes unit point mass at $y$. Only the small central atom moves. The manuscript gives the exact identities

$$W_2(\mu_a,\nu_a)^2=ab^2,$$

and

$$\|T_{\mu_a}-T_{\nu_a}\|_{L^2(\rho)}^2 =\frac b2+ab^2.$$

Thus target distance has order $a^{3/2}$, while map error has order $a^{1/2}$. Their relation is precisely a cube root. The larger map error comes from source points reassigned between destinations separated by order-one distances, not from the tiny movement of the central atom alone.

For the same example $W_1=ab$, of order $a^2$. Its map error therefore also fits a quarter-power $W_1$ obstruction. The two sharp exponents are compatible because the two target distances scale differently here.

Control finite cells without a separation assumption

For the upper bound, first take a finite target with sites $y_i$. Its potential has the form

$$\varphi_c(x)=\max_i\{x\cdot y_i+c_i\}.$$

The intercepts $c_i$ determine convex cells $P_i$ on which the map equals $y_i$. Let

$$M_i=\rho(P_i), \qquad B_i=\int_{P_i}x\,d\rho(x)$$

be each cell’s mass and first moment.

The source’s central estimate compares how moments and masses change when the intercepts vary. If dots denote these derivatives and $K\subset B(0,R)$, the claimed inequality is

$$\sum_i\frac{|\dot B_i|^2}{M_i} \le C_0R^2\sum_i\frac{|\dot M_i|^2}{M_i}.$$

Its constant stays controlled even when cells or faces become small. The proof couples uniform measures on nearby cells and controls the density of their midpoint distribution using convexity and a triangular rearrangement. Testing a concave quadratic function and summing over the partition turns that geometric comparison into the moment estimate.

Move target sites while preserving their masses

An optimal coupling between two finite targets gives a way to move their sites while retaining the coupling masses. Keeping the source-cell masses fixed determines how the intercepts must change.

The resulting linear equation is a weighted graph-Laplacian equation: neighboring cells exchange mass through shared faces. Direct inversion can be poorly conditioned when cells become tiny. The manuscript instead tests the equation in a direction chosen to exploit the moment inequality, obtaining a bound for the weighted intercept velocity without a quantitative inverse-Laplacian estimate.

After integrating the motion, this yields a potential estimate of the form

$$\|\varphi_\mu-\varphi_\nu\|_{L^2(\rho)} \le C W_2(\mu,\nu),$$

with additive constants normalized appropriately.

Turn potential control into gradient control

Potential values can be much closer than their gradients near moving cell boundaries. Convexity supplies an interpolation estimate. For a step length $h>0$, comparison with finite differences along coordinate slices gives

$$\|\nabla\varphi_\mu-\nabla\varphi_\nu\|_{L^2(\rho)}^2 \le C\left(h^{-2}\|\varphi_\mu-\varphi_\nu\|_{L^2(\rho)}^2+h\right).$$

Convexity makes every nonempty coordinate slice an interval and bounds the accumulated derivative changes along it. Boundary slices contribute the term proportional to $h$, without requiring a smooth boundary.

Insert the potential estimate and balance the terms with $h$ of order $W_2^{2/3}$. The squared map error is then of order $W_2^{2/3}$, giving the one-third bound after taking a square root.

Finally, finite approximations of arbitrary target measures and supporting inequalities for convex potentials extend the result beyond discrete targets. Uniformity of the finite-cell estimates is what makes that passage possible.

7. Scope, limitations, and open questions

Sharpness concerns one constant valid for all targets in a compact set. It does not prevent a better local rate near a regular target or a better uniform rate on a class with density and nondegeneracy bounds.

The source is fixed and uniform on a convex body. Nonuniform sources, changing sources, nonconvex supports, other transport costs and manifold settings require separate hypotheses. Dimension one is not part of the sharpness statement.

The norm is an averaged $L^2$ map error. It does not give uniform pointwise closeness of maps, continuity of discontinuous cell assignments, or a universal numerical convergence rate independent of an approximation scheme.

The theorem identifies the endpoint in this source class; understanding which weaker or different source assumptions retain it is a distinct question.

8. Sources and further reading

The Brenier, Delalande–Mérigot, Divol–Niles-Weed–Pooladian and Letrouit papers linked above locate the representation, interpolation method, finite-target predecessor and conjectured exponent. Kitagawa, Mérigot and Thibert develop the semi-discrete cell calculus and graph-Laplacian viewpoint used in this area.

Optional background on Brenier’s theorem and Wasserstein distance complements the definitions. The manuscript below supplies the claimed uniform endpoint and matching three-atom example.

Source manuscripts

• M1. OpenAI, Sharp One-Third Stability of Brenier Maps (September 25, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.