# Tight stability bounds for entropic Brenier maps

Vincent Divol Jonathan Niles-Weed Aram-Alexandre Pooladian CEREMADE,
Université Paris-Dauphine - PSL<EMAIL_ADDRESS>Institute of
Mathematical Sciences and Center for Data Science, New York University.
<EMAIL_ADDRESS>for Data Science, New York University. aram-
<EMAIL_ADDRESS>
###### Abstract

Entropic Brenier maps are regularized analogues of Brenier maps (optimal
transport maps) which converge to Brenier maps as the regularization parameter
shrinks. In this work, we prove quantitative stability bounds between entropic
Brenier maps under variations of the target measure. In particular, when all
measures have bounded support, we establish the optimal Lipschitz constant for
the mapping from probability measures to entropic Brenier maps. This provides
an exponential improvement to a result of Carlier, Chizat, and Laborde (2024).
As an application, we prove near-optimal bounds for the stability of semi-
discrete _unregularized_ Brenier maps for a family of discrete target
measures.

##  1 Introduction

The theory of optimal transport defines a geometry over probability measures
via the _$2$ -Wasserstein distance_: for a source measure $\rho$ and a target
measure $\mu$ with finite second moments, their Wasserstein distance is given
by

$\displaystyle
W_{2}^{2}(\rho,\mu)\coloneqq\min_{T:T_{\sharp}\rho=\mu}\int\|x-T(x)\|^{2}\,\mathrm{d}\rho(x)\,,$
(1)

where the constraint $T_{\sharp}\rho=\mu$ means that for $X\sim\rho$,
$T(X)\sim\mu$, i.e., $T$ is a transport map. The minimizer to (1), when it
exists, is called an optimal transport map, which we denote by $T_{0}^{\mu}$.
A seminal result by Brenier (1991) states that a unique optimal transport map
between $\rho$ and $\mu$ exists whenever $\rho$ has a density, and moreover
$T_{0}^{\mu}=\nabla\varphi_{0}^{\mu}$, where $\varphi_{0}^{\mu}$ is some
convex function. We will henceforth refer to optimal transport maps as
_Brenier maps_ , and the corresponding convex functions that generate them as
_Brenier potentials_.

A long-standing question in the optimal transport community is the following:
is the mapping $\mu\mapsto T_{0}^{\mu}$ Hölder continuous with respect to the
$2$-Wasserstein distance? In other words, do there exist constants $C,\beta>0$
such that for all probability measures $\mu$, $\nu$ with finite second
moments,

$\displaystyle\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}\leq
CW^{\beta}_{2}(\mu,\nu)\,?$ (2)

Since the inequality
$W_{2}(\mu,\nu)\leq\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}$ always holds,
(2) would imply that the mapping $\mu\mapsto T_{0}^{\mu}$ is a bi-Hölder
embedding of the Wasserstein space into $L^{2}(\rho)$. We call such an
inequality a _stability bound_.

The unique structure of the one-dimensional optimal transport problem shows
that when $\rho$, $\mu$, and $\nu$ are probability measures on $\mathbb{R}$,
the bound (2) holds with $C=\beta=1$—that is, the mapping $\mu\mapsto
T_{0}^{\mu}$ is an isometry (see, e.g., Panaretos and Zemel, 2020, Chapter 2).
On the other hand, Andoni et al. (2015) showed that if $d\geq 3$, then (2)
cannot hold uniformly over all probability measures $\mu$ and $\nu$ on
$\mathbb{R}^{d}$ with finite second moment. In fact, their main statement is
significantly stronger and rules out the possibility of embedding the
Wasserstein space into any $L^{p}$ space, even in a very weak sense.
Nevertheless, as we describe further below, a stability bound such as (2) can
hold if further conditions are imposed on $\mu$ and $\nu$, for instance, if
they are compactly supported.

An early investigation in this direction is due to Gigli (2011), who showed
that even when $\mu$ and $\nu$ are compactly supported, the exponent in (2)
cannot be better than $\beta=\tfrac{1}{2}$. However, in the same paper, the
author reports a simple proof due to Ambrosio that shows that if one of the
Brenier maps, say $T_{0}^{\nu}$, is $\Lambda$-Lipschitz, then
$\beta=\tfrac{1}{2}$ is achievable, with $C=2\sqrt{\Lambda R}$, where $R$ is
the diameter of the support of $\rho$; see also Mérigot et al. (2020, Theorem
2.3) for a precise statement and proof of this result. More recently, Manole
et al. (2021) showed that if $T_{0}^{\nu}$ is $\Lambda$-Lipschitz and its
inverse is $1/\lambda$-Lipschitz, then $\beta=1$ is achievable, with constant
$C=\sqrt{\Lambda/\lambda}$.

Though these positive results are encouraging, requiring _a priori_ smoothness
bounds on one of the two Brenier potentials excludes many cases of practical
interest, for instance, the case of discontinuous Brenier maps. Such maps
arise commonly in applications of optimal transport to machine learning, where
it is natural to consider probability measures that lie on a union of
manifolds of different intrinsic dimension (Brown et al., 2022). There has
therefore been significant recent interest in obtaining stability bounds
without such assumptions; see Mérigot et al. (2020), Berman (2021), Delalande
and Mérigot (2023). The results of Delalande and Mérigot (2023) are the most
recent. They show that if $\rho$ has a (uniformly upper and lower bounded)
density supported on a convex set $\mathcal{X}$, with $\mu$ and $\nu$ also
supported on a compact set $\mathcal{Y}$, then

$\displaystyle\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}\leq
C_{d,\mathcal{X},\mathcal{Y},\rho}W^{1/6}_{2}(\mu,\nu)\,.$

In fact, the authors prove this bound for the $W_{1}$ distance. Their proof
technique relies on applications of the Brascamp–Lieb and Prekopa–Leindler
inequalities.

The goal of this paper is to prove analogous stability bounds for _entropic
Brenier maps_. Entropic Brenier maps are defined as barycentric projections of
entropic optimal couplings between $\rho$ and $\mu$ (resp. $\rho$ and $\nu$),
written in short-hand as $T_{\varepsilon}^{\mu}$ (resp.
$T_{\varepsilon}^{\nu}$), where $\varepsilon>0$ is a regularization parameter
(we defer detailed background to Section 2.2). Our motivation for studying
such maps is twofold. First, entropic Brenier maps are prominent in
applications for their statistical and computational properties. These maps
arise directly from the output of _Sinkhorn’s algorithm_ , a very popular
computational approximation to optimal transport (Peyré and Cuturi, 2019).
Moreover, Pooladian and Niles-Weed (2021) advocated for their use as a
statistical tool, and analyzed the entropic Brenier map as an estimator for
the unregularized Brenier map on the basis of i.i.d. data from $\rho$ and
$\mu$. Second, entropic optimal transport is a natural smoothed analogue to
the optimal transport problem, and it is reasonable to hope that techniques
developed for entropic optimal transport can give insights into the structure
of the unregularized problem.

Despite the importance of entropic Brenier maps, much less is known about
their stability properties. The first result in this area is due to Carlier et
al. (2024), who showed that if $\rho$, $\mu$, and $\nu$ are compactly
supported, then

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq
C_{\varepsilon}W_{2}(\mu,\nu)\,,$ (3)

where $C_{\varepsilon}$ is a constant that grows exponentially as
$\varepsilon$ tends to zero.

This striking result reveals that entropic Brenier maps automatically enjoy
better stability properties than unregularized Brenier maps when
$\varepsilon>0$. However, if (3) is to be used to extract either practical
bounds for entropic Brenier maps or insights about unregularized Brenier maps
in the $\varepsilon\to 0$ limit, it is crucial to obtain sharp bounds on the
constant $C_{\varepsilon}$.

### Contributions

The goal of this paper is to improve the Lipschitz constant for the embedding
$\mu\mapsto T_{\varepsilon}^{\mu}$ as a function of $\varepsilon$. Our main
theorem is technical, but it readily implies results in the following three
scenarios of interest.

First, if the source and target measures are merely supported in the Euclidean
ball of radius $R$, then

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq\bigl{(}1+2R^{2}/\varepsilon\bigr{)}W_{2}(\mu,\nu)\,,$

see Corollary 3.3. We stress that none of the measures here require densities,
and so, a priori, Brenier maps may not exist, while their entropic
counterparts do. Moreover, up to universal constants, we show that this bound
is tight; see Remark 3.4. This is an exponential improvement on the bounds
provided by Carlier et al. (2024).

As in the unregularized case, the preceding bounds can be improved under
smoothness assumptions on the entropic Brenier potentials. Such assumptions
are arguably more reasonable than in the unregularized case, since it is
sometimes possible to obtain _a priori_ smoothness bounds for entropic Brenier
potentials via elementary tools (see, e.g., Chewi and Pooladian, 2023). If one
of the entropic Brenier maps, say $T_{\varepsilon}^{\nu}$, is
$\Lambda$-Lipschitz, we show that the previous bound can be improved to

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq\bigl{(}1+2\sqrt{{R\Lambda/\varepsilon}}\bigr{)}W_{2}(\mu,\nu)\,,$

Going further, if the backward entropic Brenier map $S_{\varepsilon}^{\nu}$
(see Section 2.2 for a precise definition) is $1/\lambda$-Lipschitz, then the
bound becomes independent of the regularization parameter:

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq\bigl{(}1+2\sqrt{{\Lambda/\lambda}}\bigr{)}W_{2}(\mu,\nu)\,.$

See Corollary 3.5 for these last two results. In particular, up to constants,
this result is analogous to the stability bound established by Manole et al.
(2021).

As a novel application, we turn to the _semi-discrete_ setting of optimal
transport, where $\mu$ and $\nu$ are both supported on finitely many atoms and
$\rho$ has a sufficiently well-behaved density. In this setting, we partially
close the gap left by Gigli and others, where we prove that

$\displaystyle\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}\lesssim
W^{1/3}_{2}(\mu,\nu)\,,$ (4)

where $\mu,\nu$ satisfy appropriate regularity conditions, as does the source
measure $\rho$, and the suppressed constant depends on these regularity
assumptions. While our results do not allow for arbitrary discrete measures,
they hold for a wide class of discrete measures and do not require the support
of the atoms to be the same. The proof starts from the following application
of the triangle inequality

$\displaystyle\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}\leq\|T_{0}^{\mu}-T_{\varepsilon}^{\mu}\|_{L^{2}(\rho)}+\|T_{0}^{\nu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}+C_{\varepsilon}W_{2}(\mu,\nu)\,.$

Under appropriate assumptions on $\rho$ and the two discrete measures $\mu$
and $\nu$, we are able to control the first two terms using existing
techniques, and the third term can be controlled via Corollary 3.3. Balancing
the resulting terms as a function of $\varepsilon$, we obtain the final bound
that appears in (4). Our identification of the sharp constant
$C_{\varepsilon}$ is crucial to obtaining the result. See Section 4 for more
details.

##  2 Background on optimal transport

###  2.1 Optimal transport

Let $\mathcal{P}_{2}$ be the space of probability measures on $\mathbb{R}^{d}$
with finite second moment. For two probability measures
$\rho,\nu\in\mathcal{P}_{2}$, we define the (squared) $2$-Wasserstein distance
by

$\displaystyle\tfrac{1}{2}W_{2}^{2}(\rho,\nu):=\inf_{\pi\in\Gamma(\rho,\nu)}\iint\tfrac{1}{2}\|x-z\|^{2}\,\mathrm{d}\pi(x,z)\,,$
(5)

where $\Gamma(\rho,\nu)$ is the set of joint measures with left-marginal
$\rho$ and right-marginal $\nu$; see Villani (2009), Santambrogio (2015) for
more information. When $\rho,\nu$ have finite second moments, solutions to (5)
always exist. When $\rho$ has a density, (5) has a unique minimizer,
$\pi_{0}$, called the the _optimal transport coupling_.

From (5), one can derive the _dual formulation_ of the 2-Wasserstein distance

$\displaystyle\tfrac{1}{2}W_{2}^{2}(\rho,\nu)$
$\displaystyle=\tfrac{1}{2}M_{2}(\rho+\nu)-\inf_{\varphi\in
L^{1}(\rho)}\int\varphi\,\mathrm{d}\rho+\int\varphi^{*}\,\mathrm{d}\nu\,,$ (6)

where $M_{2}(P):=\int\|x\|^{2}\,\mathrm{d}P(x)$ for a measure $P$, and
$\varphi^{*}$ is the convex conjugate of $\varphi$, defined for
$x\in\mathbb{R}^{d}$ by $\varphi^{*}(x)\coloneqq\sup_{y}\\{\langle
x,y\rangle-\varphi(y)\\}.$

Similar to the primal problem, whenever $\rho$ has a density, the dual problem
(6) admits a minimizer $\varphi_{0}$, which can be shown to be a convex
function. In fact, we obtain a pair of _Brenier potentials_ , written

$\displaystyle(\varphi_{0},\psi_{0})\coloneqq(\varphi_{0},\varphi_{0}^{*})=(\psi_{0}^{*},\psi_{0})\,.$

Furthermore, the pair $(\varphi_{0},\psi_{0})$ is unique up to additive
constants $(\varphi_{0}+c,\psi_{0}-c)$.

Recalling (1), the following theorem unifies these three formulations of
optimal transport under the squared-Euclidean cost.

######  Theorem 2.1 (Brenier, 1991).

Let $\rho,\nu\in\mathcal{P}_{2}$. Assume that $\rho$ has a density with
respect to the Lebesgue measure. Then,

  1. 1.

the solution to (1) exists and is of the form
$T_{0}=\nabla\varphi_{0}=\nabla\psi^{*}_{0}$, with $\varphi_{0}$ convex being
a solution to (6).

  2. 2.

$\pi_{0}$ is also uniquely defined as

$\,\mathrm{d}\pi_{0}(x,y)=\,\mathrm{d}\rho(x)\delta_{\\{\nabla\varphi_{0}(x)\\}}(y)\,.$

If $\nu$ also has a density, then the _backward_ Brenier map, from $\nu$ to
$\rho$, is given by

$\displaystyle(T_{0})^{-1}=\nabla\psi_{0}=\nabla\varphi_{0}^{*}\,.$

To be concise, we simply write $\varphi_{0}^{\nu}$, $\psi_{0}^{\nu}$ and
$T_{0}^{\nu}$ to refer to the optimal transport quantities associated to the
problem $W_{2}^{2}(\rho,\nu)$ (and similarly for $W_{2}^{2}(\rho,\mu)$).

###  2.2 Entropic optimal transport

For two probability measures $\rho,\nu\in\mathcal{P}_{2}$, the entropic
optimal transport objective (Cuturi, 2013) is defined as

$\displaystyle\tfrac{1}{2}W_{2,\varepsilon}^{2}(\rho,\nu)\coloneqq\min_{\pi\in\Gamma(\rho,\nu)}\iint\tfrac{1}{2}\|x-z\|^{2}\,\mathrm{d}\pi(x,z)+\varepsilon\text{KL}(\pi\|\rho\otimes\nu)\,,$
(7)

for some $\varepsilon>0$, and $\text{KL}(\pi\|\rho\otimes\nu)$ is the
Kullback–Leibler divergence, defined as

$\displaystyle\text{KL}(\pi\|\rho\otimes\nu)\coloneqq\int\log\left(\frac{\,\mathrm{d}\pi}{\,\mathrm{d}\rho\otimes\,\mathrm{d}\nu}\right)\,\mathrm{d}\pi$

when $\pi$ is absolutely continuous with respect to $\rho\otimes\nu$, and
$+\infty$ otherwise. Note that due to the regularization term, the problem is
strictly convex with a unique minimizer $\pi_{\varepsilon}^{\nu}$, the
_optimal entropic (transport) coupling_.***Since $\rho$ is fixed throughout,
we use the superscript $\nu$ (respectively, $\mu$) to indicate objects that
correspond to the entropic optimal transport problem between $\rho$ and $\nu$
(respectively, $\rho$ and $\mu$).

The entropic optimal transport problem also admits a dual formulation (see,
e.g., Genevay, 2019):

$\displaystyle\tfrac{1}{2}W_{2,\varepsilon}^{2}(\rho,\nu)\coloneqq\tfrac{1}{2}M_{2}(\rho+\nu)-\min_{\varphi\in
L^{1}(\rho)}\int\varphi\,\mathrm{d}\rho+\int\Phi_{\varepsilon}^{\rho}[\varphi]\,\mathrm{d}\nu\,,$
(8)

where $\Phi_{\varepsilon}^{\nu}$ is the following operator

$\displaystyle\forall z\in\mathbb{R}^{d},\
\Phi_{\varepsilon}^{\rho}[\varphi](z)\coloneqq\varepsilon\log\int e^{(\langle
x,z\rangle-\varphi(x))/\varepsilon}\,\mathrm{d}\rho(x)\,,$

which should be thought of as the entropic analogue to the convex conjugate
operator. Indeed, notice that as $\varepsilon\to 0$,
$\Phi_{\varepsilon}^{\rho}[\varphi](z)$ converges to the $\rho$-essential
supremum of the function $x\mapsto\langle x,z\rangle-\varphi(x)$. We write the
minimizer to (8) as $\varphi_{\varepsilon}^{\nu}$, from which we obtain the
minimizing pair of _entropic Brenier potentials_

$\displaystyle(\varphi_{\varepsilon}^{\nu},\psi_{\varepsilon}^{\nu})\coloneqq(\varphi_{\varepsilon}^{\nu},\Phi_{\varepsilon}^{\rho}[\varphi_{\varepsilon}^{\nu}])=(\Phi_{\varepsilon}^{\nu}[\psi_{\varepsilon}^{\nu}],\psi_{\varepsilon}^{\nu})\,,$

where $\Phi_{\varepsilon}^{\nu}$ is defined analogously to
$\Phi_{\varepsilon}^{\rho}$. Again, this pair is unique up to constant shifts.

Moreover, by the dual optimality conditions, we can define versions of the
entropic Brenier potentials taking values in the extended reals, for all
$x\in\mathbb{R}^{d}$ and $z\in\mathbb{R}^{d}$, respectively. Thus, we freely
write

$\begin{split}&\varphi_{\varepsilon}^{\nu}(x)\coloneqq\varepsilon\log\int
e^{(\langle
x,z\rangle-\psi_{\varepsilon}^{\nu}(z))/\varepsilon}\,\mathrm{d}\nu(z)\quad(x\in\mathbb{R}^{d})\\\
&\psi_{\varepsilon}^{\nu}(z)\coloneqq\varepsilon\log\int e^{(\langle
x,z\rangle-\varphi_{\varepsilon}^{\nu}(x))/\varepsilon}\,\mathrm{d}\rho(x)\quad(z\in\mathbb{R}^{d})\,.\end{split}$
(9)

See Mena and Niles-Weed (2019), Nutz and Wiesel (2021) for more details. Note
that
$\varphi_{\varepsilon}^{\nu}:\mathbb{R}^{d}\to\mathbb{R}\cup\\{+\infty\\}$
(resp. $\psi_{\varepsilon}^{\nu}$) is a convex function which is analytic on
the interior of its domain $\operatorname{dom}(\varphi_{\varepsilon}^{\nu})$
(resp. $\operatorname{dom}(\psi_{\varepsilon}^{\nu})$),

An important feature of the entropic optimal transport problem is that the
optimal solutions to (7) and (8) satisfy the following primal-dual
relationship (Csiszár, 1975):

$\displaystyle\,\mathrm{d}\pi_{\varepsilon}^{\nu}(x,z)\coloneqq\gamma_{\varepsilon}^{\nu}(x,z)\,\mathrm{d}\rho(x)\,\mathrm{d}\nu(z)\coloneqq
e^{(\langle
x,z\rangle-\psi_{\varepsilon}^{\nu}(z)-\varphi_{\varepsilon}^{\nu}(x))/\varepsilon}\,\mathrm{d}\rho(x)\,\mathrm{d}\nu(z)\,.$

Concretely, $\gamma_{\varepsilon}^{\nu}$, the density of
$\pi_{\varepsilon}^{\nu}$ with respect to $\rho\otimes\nu$, can be written
explicitly in terms of the entropic Brenier potentials
$(\varphi_{\varepsilon}^{\nu},\psi_{\varepsilon}^{\nu})$.

Let $(X,Z)$ be a pair of random variables with distribution
$\pi_{\varepsilon}^{\nu}$. For a given
$x\in\operatorname{dom}(\varphi_{\varepsilon}^{\nu})$, we abuse notation and
define the conditional probability of $Z|X=x$ as

$\displaystyle\,\mathrm{d}\pi_{\varepsilon}^{\nu}(z|x)=e^{(\langle
x,z\rangle-\varphi_{\varepsilon}^{\nu}(x)-\psi_{\varepsilon}^{\nu}(z))/\varepsilon}\,\mathrm{d}\nu(z)\,.$

Similarly we denote
$\pi_{\varepsilon}^{\nu}(\cdot|z)\coloneqq\pi_{\varepsilon}^{\nu}(\cdot|Z=z)$
whenever $z\in\operatorname{dom}(\psi_{\varepsilon}^{\nu})$. Likewise, if
$(X,Y)$ are distributed according to the optimal entropic coupling
$\pi_{\varepsilon}^{\mu}$ between $\rho$ and $\mu$, we will write
$\pi_{\varepsilon}^{\mu}(\cdot|x)$ and $\pi_{\varepsilon}^{\mu}(\cdot|y)$ for
the conditional distributions of $Y|X=x$ and $X|Y=y$, respectively. We will
adopt the convention throughout that $X$, $Y$, and $Z$ always refer to random
variables with marginal distributions $\rho$, $\mu$, and $\nu$, respectively.

Following e.g., Pooladian and Niles-Weed (2021), Rigollet and Stromme (2022),
we define, respectively, the _forward_ and _backward entropic Brenier maps_
from $\rho$ to $\nu$ to be barycentric projections of
$\pi_{\varepsilon}^{\nu}$ (Ambrosio et al., 2008, Definition 5.4.2): for
$x\in\operatorname{dom}(\varphi_{\varepsilon}^{\nu})$ and
$z\in\operatorname{dom}(\psi_{\varepsilon}^{\nu})$, we define

$T_{\varepsilon}^{\nu}(x)\coloneqq\int
z\,\mathrm{d}\pi_{\varepsilon}^{\nu}(z|x)\,,\quad
S_{\varepsilon}^{\nu}(z)\coloneqq\int
x\,\mathrm{d}\pi_{\varepsilon}^{\nu}(x|z)\,$

whenever the integrals are well-defined. Unlike the unregularized case,
$(S_{\varepsilon}^{\nu})^{-1}\neq T_{\varepsilon}^{\nu}$. Note that by
Jensen’s inequality, $T_{\varepsilon}^{\nu}\in L^{2}(\rho)$ with
$\|T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}^{2}\leq M_{2}(\nu)$ (resp.
$S_{\varepsilon}^{\nu}\in L^{2}(\nu)$ with
$\|S_{\varepsilon}^{\nu}\|_{L^{2}(\nu)}^{2}\leq M_{2}(\rho)$). Also note that
by the dominated convergence theorem, the gradient of
$\varphi_{\varepsilon}^{\nu}$ (resp. $\psi_{\varepsilon}^{\nu}$) from (9) has
a natural interpretation as the forward (resp. backward) entropic Brenier map:
whenever $x$ is in the interior of
$\operatorname{dom}(\varphi_{\varepsilon}^{\nu})$ and $z$ is in the interior
of $\operatorname{dom}(\psi_{\varepsilon}^{\nu})$,

$\nabla\varphi_{\varepsilon}^{\nu}(x)=T_{\varepsilon}^{\nu}(x)\,,\quad\nabla\psi_{\varepsilon}^{\nu}(z)=S_{\varepsilon}^{\nu}(z)\,.$
(10)

Under the same condition, a similar expression holds for the Hessians of the
entropic Brenier potentials (see, e.g., Chewi and Pooladian, 2023, Lemma 1):

$\displaystyle\nabla^{2}\varphi_{\varepsilon}^{\nu}(x)=\varepsilon^{-1}\text{Cov}_{\pi_{\varepsilon}^{\nu}}(Z|X=x)\,,\quad\nabla^{2}\psi_{\varepsilon}^{\nu}(z)=\varepsilon^{-1}\text{Cov}_{\pi_{\varepsilon}^{\nu}}(X|Z=z)\,.$
(11)

Throughout, we will write
$(\varphi_{\varepsilon}^{\nu},\psi_{\varepsilon}^{\nu})$ for the entropic
Brenier potentials associated to
$\tfrac{1}{2}W_{2,\varepsilon}^{2}(\rho,\nu)$, $T_{\varepsilon}^{\nu}$ for the
forward entropic Brenier map, and $S_{\varepsilon}^{\nu}$ for the backward
entropic Brenier map (and similarly for
$\tfrac{1}{2}W_{2,\varepsilon}^{2}(\rho,\mu)$).

#### Related work in entropic optimal transport

##### Fixed regularization.

The initial motivation for studying (7) in the machine learning literature was
its significant computational benefits compared to (5) (Cuturi, 2013,
Altschuler et al., 2017). As a result, the study of entropic objects for a
fixed $\varepsilon>0$ regularization parameter has been of great interest in a
number of fields. For example, del Barrio et al. (2022), Goldfeld et al.
(2022), Gonzalez-Sanz et al. (2022) studied statistical limit theorems for
entropic optimal transport. Greco et al. (2023), Conforti et al. (2023), Nutz
and Wiesel (2023) study the convergence of Sinkhorn’s algorithm to the optimal
Brenier potentials at the population level. The works by Masud et al. (2021),
Pooladian et al. (2022), Rigollet and Stromme (2022), Stromme (2023), Klein et
al. (2023), Werenski et al. (2023) studied additional computational or
statistical properties of entropic Brenier maps. As previously mentioned,
Carlier et al. (2024) initiated the study of the stability properties of
entropic Brenier maps under variations of the target measure, though their
techniques differ significantly from ours.

##### Vanishing regularization.

Theoretical properties of entropic optimal transport for vanishing
regularization parameter are widely studied in both statistical and
theoretical works. For example, convergence of the regularized to
unregularized couplings was studied by Léonard (2012), Carlier et al. (2017),
Bernton et al. (2022), Ghosal et al. (2022), and convergence of the transport
costs by Chizat et al. (2020), Conforti and Tamanini (2021), Eckstein and Nutz
(2023), Pal (2024). Nutz and Wiesel (2021) established convergence of the
entropic to non-entropic Brenier potentials under minimal assumptions; this
convergence was improved in the case of semi-discrete optimal transport by
Altschuler et al. (2022) and Delalande (2022). Chewi and Pooladian (2023)
established a short proof of Caffarelli’s contraction theorem (Caffarelli,
2000) via covariance inequalities and entropic optimal transport, which was
subsequently generalized by Conforti (2022). Statistical convergence of
entropic Brenier maps to unregularized Brenier maps was established by
Pooladian and Niles-Weed (2021), Pooladian et al. (2023), the latter paper
focusing on the semi-discrete setting.

###  2.3 A transport inequality for conditional entropic couplings

At the core of our approach is the use of a specific transport inequality
which has been developed for other purposes in the study of sampling and
functional inequalities (Anari et al., 2021b, a, Chen and Eldan, 2022,
Bauerschmidt et al., 2023). We refer to Bauerschmidt et al. (2023, Section
3.7) for more details, and briefly overview the necessary inequalities and
notation here.

Let $q\in\mathcal{P}_{2}$ be a probability measure with finite moment
generating function whose covariance is denoted by $\text{Cov}(q)$. For
$h\in\mathbb{R}^{d}$, we define the tilt $\mathcal{T}_{h}q$ of $q$ as the
probability measure satisfying

$\displaystyle\forall
z\in\mathbb{R}^{d},\quad\frac{\,\mathrm{d}\mathcal{T}_{h}q}{\,\mathrm{d}q}(z)\coloneqq\frac{\exp(\langle
h,z\rangle)}{\mathbb{E}_{Z\sim q}[\exp(\langle h,Z\rangle)]}\,.$

We say that $q$ is _tilt-stable_ †††This name is not standard, but we
introduce it here because the standard name for this concept (entropic
stability) is likely to cause confusion in the context of our main results. if
for all $h\in\mathbb{R}^{d}$, $\text{Cov}(\mathcal{T}_{h}q)\preceq
C_{\mathsf{T}}I$ for some $C_{\mathsf{T}}>0$. If $q$ is tilt-stable with
constant $C_{\mathsf{T}}$, then for all probability measures
$p\in\mathcal{P}_{2}$,

$\displaystyle\|\mathbb{E}_{p}[X]-\mathbb{E}_{q}[X]\|^{2}\leq
2C_{\mathsf{T}}\text{KL}(p\|q)\,,$

see (Bauerschmidt et al., 2023, Lemma 3.21).

Our main observation is that conditional entropic couplings are tilt-stable,
with a constant that can be written in terms of the entropic Brenier
potentials. For an entropic potential $\varphi_{\varepsilon}^{\nu}$ whose
domain is all of $\mathbb{R}^{d}$, we write

$H_{\max}(\varphi_{\varepsilon}^{\nu})\coloneqq\sup_{u\in\mathbb{R}^{d}}\|\text{Cov}_{\pi_{\varepsilon}^{\nu}}(Z|X=u)\|_{\text{op}}=\varepsilon\sup_{u\in\mathbb{R}^{d}}\|\nabla^{2}\varphi_{\varepsilon}^{\nu}(u)\|_{\text{op}}\,,$
(12)

and define $H_{\max}(\psi_{\varepsilon}^{\nu})$ analogously. The second
equality in (12) is justified by the fact that the identity (11) holds
everywhere when
$\operatorname{dom}(\varphi_{\varepsilon}^{\nu})=\mathbb{R}^{d}$. If either
potential is not finite on all of $\mathbb{R}^{d}$, we adopt the convention
that $H_{\max}=+\infty$.

######  Lemma 2.2.

Let $x\in\mathbb{R}^{d}$ and let $\pi_{\varepsilon}^{\nu}(\cdot|x)$ be a
conditional entropic coupling between two probability measures
$\rho,\nu\in\mathcal{P}_{2}$. Assume that
$\operatorname{dom}(\varphi_{\varepsilon}^{\nu})=\mathbb{R}^{d}$. Then for any
$h\in\mathbb{R}^{d}$,

$\displaystyle\mathcal{T}_{h}\pi_{\varepsilon}^{\nu}(\cdot|x)=\pi_{\varepsilon}^{\nu}(\cdot|x+\varepsilon
h)\,.$

###### Proof.

We temporarily omit the superscript in $\nu$ for ease of reading. Note that

$\displaystyle\mathbb{E}_{Z\sim\pi_{\varepsilon}(\cdot|x)}[e^{\langle
h,Z\rangle}]=e^{-\varphi_{\varepsilon}(x)/\varepsilon}\int e^{(\langle
x+\varepsilon
h,z\rangle-\psi_{\varepsilon}(z))/\varepsilon}\,\mathrm{d}\nu(z)=e^{(\varphi_{\varepsilon}(x+\varepsilon
h)-\varphi_{\varepsilon}(x))/\varepsilon}\,.$

From this we can conclude, since for all $z\in\mathbb{R}^{d}$

$\displaystyle\frac{\,\mathrm{d}\mathcal{T}_{h}\pi_{\varepsilon}(\cdot|x)}{\,\mathrm{d}\nu}(z)$
$\displaystyle=e^{(\langle x+\varepsilon
h,z\rangle-\varphi_{\varepsilon}(x)-\psi_{\varepsilon}(z))/\varepsilon}e^{(\varphi_{\varepsilon}(x)-\varphi_{\varepsilon}(x+\varepsilon
h))/\varepsilon}$ $\displaystyle=e^{(\langle x+\varepsilon
h,z\rangle-\varphi_{\varepsilon}(x+\varepsilon
h)-\psi_{\varepsilon}(z))/\varepsilon}$
$\displaystyle=\frac{\,\mathrm{d}\pi_{\varepsilon}(\cdot|x+\varepsilon
h)}{\,\mathrm{d}\nu}(z)\,.\qed$

######  Corollary 2.3.

The conditional entropic coupling $\pi_{\varepsilon}^{\nu}(\cdot|x)$ (resp.
$\pi_{\varepsilon}^{\nu}(\cdot|z)$) is tilt-stable with constant
$H_{\max}(\varphi_{\varepsilon}^{\nu})$ (resp.
$H_{\max}(\psi_{\varepsilon}^{\nu})$).

###### Proof.

If the domain of $\varphi_{\varepsilon}^{\nu}$ is not $\mathbb{R}^{d}$, then
$H_{\max}(\varphi_{\varepsilon}^{\nu})=+\infty$, and the proposition is
vacuous. Otherwise, fix $x\in\mathbb{R}^{d}$. By definition of tilt stability,
it suffices to compute an upper bound on the covariance of
$\mathcal{T}_{h}\pi^{\nu}_{\varepsilon}(\cdot|x)$ which holds uniformly over
all tilts $h\in\mathbb{R}^{d}$. This follows by direct computation, as Lemma
2.2 and (12) imply that

$\displaystyle\text{Cov}(\mathcal{T}_{h}\pi_{\varepsilon}^{\nu}(\cdot|x))=\text{Cov}(\pi_{\varepsilon}^{\nu}(\cdot|x+\varepsilon
h))\preceq H_{\max}(\varphi_{\varepsilon}^{\nu})I\,,$

where the last inequality holds by taking the supremum over both $x$ and $h$
arguments. Note that the argument is symmetric for the other conditional
entropic coupling. ∎

##  3 Main results

We now present our general stability result for entropic Brenier maps.

######  Theorem 3.1 (Stability of entropic Brenier maps).

Suppose $\rho,\mu,\nu$ have finite second moment. Then

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq\Bigl{(}1+\frac{2(H_{\max}(\varphi_{\varepsilon}^{\nu})H_{\max}(\psi_{\varepsilon}^{\nu}))^{1/2}}{\varepsilon}\Bigr{)}W_{2}(\mu,\nu)\,.$

######  Remark 3.2.

Note that if the potentials
$(\varphi_{\varepsilon}^{\nu},\psi_{\varepsilon}^{\nu})$ are not finite
everywhere, the quantities $H_{\max}(\varphi^{\nu}_{\varepsilon})$ and
$H_{\max}(\psi^{\nu}_{\varepsilon})$ are infinite by convention, and the
inequality becomes vacuous. The potentials are finite everywhere whenever
$\rho$ and $\nu$ have moment-generating functions that are finite everywhere
(including the important case of bounded supports), but also when $\rho$ and
$\nu$ have support equal to $\mathbb{R}^{d}$, without additional tail
assumptions.

From here, we can prove the results highlighted in the introduction as special
cases.

######  Corollary 3.3 (Entropic stability for bounded measures).

Suppose $\rho$ and $\nu$ are supported in $B(0;R)$, and $\mu$ has finite
second moment. Then

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq\Bigl{(}1+\frac{2R^{2}}{\varepsilon}\Bigr{)}W_{2}(\mu,\nu)\,.$

###### Proof.

Since $\rho$ and $\nu$ are supported in $B(0;R)$, (11) implies that both
$H_{\max}(\varphi_{\varepsilon}^{\nu})$ and
$H_{\max}(\psi_{\varepsilon}^{\nu})$ are smaller than $R^{2}$, which completes
the proof. ∎

Note that Corollary 3.3 is _entirely general_ in its requirements, and does
not rely on smoothness of maps, nor do any of the measures require densities.

######  Remark 3.4 (Tightness of Corollary 3.3).

We now demonstrate that Corollary 3.3 is tight for general bounded probability
measures. Fix $R>0$, and let
$p_{\theta}\coloneqq\tfrac{1}{2}\delta_{Re_{\theta}}+\tfrac{1}{2}\delta_{-Re_{\theta}}$
with $e_{\theta}=(\cos(\theta),\sin(\theta))$, for
$\theta\in[0,\tfrac{\pi}{2}]$. Let $\rho\coloneqq p_{\pi/2}$ and
$\varepsilon>0$, and let $\pi_{\varepsilon}^{\theta}$ denote the entropic
optimal coupling between $\rho$ and $p_{\theta}$ for
$\theta\in[0,\tfrac{\pi}{2})$. One can deduce that the optimal entropic
coupling is symmetric for any such $\theta$:

$\displaystyle\pi_{\varepsilon}^{\theta}(x,y)=\pi_{\varepsilon}^{\theta}(-x,-y)\,,$

Following the calculations in Altschuler et al. (2022, Section 3), one can
choose
$\psi_{\varepsilon}^{\theta}(e_{\theta})=\psi_{\varepsilon}^{\theta}(-e_{\theta})=0$,
so that by (9), for all $x\in\mathbb{R}^{d}$,

$\displaystyle\varphi_{\varepsilon}^{\theta}(x)=\varepsilon\log\bigl{(}\tfrac{1}{2}e^{R\langle
x,e_{\theta}\rangle/\varepsilon}+\tfrac{1}{2}e^{R\langle
x,-e_{\theta}\rangle/\varepsilon}\bigr{)}\,.$

Then we compute

$\displaystyle
T_{\varepsilon}^{\theta}(x)=Re_{\theta}\bigl{(}\pi_{\varepsilon}^{\theta}(x,e_{\theta})-\pi_{\varepsilon}^{\theta}(x,-e_{\theta})\bigr{)}=Re_{\theta}\tanh({R\langle
x,e_{\theta}\rangle}/\varepsilon)\,.$

Let $\mu\coloneqq p_{0}$ and $\nu\coloneqq p_{\theta}$. Following the above
calculations, it is clear that

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}=R\|e_{\theta}\|\sqrt{\tanh^{2}(R\sin(\theta)/\varepsilon)}=\frac{R^{2}\theta}{\varepsilon}+O(\theta^{4})\,.$

It is also easy to verify that $W_{2}(p_{0},p_{\theta})\asymp\theta$, since
the optimal transport map from $p_{0}$ to $p_{\theta}$ is the standard
$2\times 2$ rotation matrix acting on the dirac masses. This example shows
that for $\theta$ small the dependence $R^{2}\varepsilon^{-1}$ in Corollary
3.3 is tight.

The following example provides the entropic analogue of Theorem 6 from Manole
et al. (2021); their result is formally recovered in the $\varepsilon\to 0$
limit.

######  Corollary 3.5 (Improved stability under smoothness).

Suppose $T_{\varepsilon}^{\nu}$ is uniformly $\Lambda$-Lipschitz. If $\rho$ is
supported in $B(0;R)$, then

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq\bigl{(}1+2\sqrt{\Lambda
R^{2}/\varepsilon}\bigr{)}W_{2}(\mu,\nu)\,.$

If instead $S_{\varepsilon}^{\nu}$ is uniformly $1/\lambda$-Lipschitz, then

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq(1+2\sqrt{\Lambda/\lambda})W_{2}(\mu,\nu)\,.$

###### Proof.

The first claim follows from the bounds
$H_{\max}(\varphi_{\varepsilon}^{\nu})\leq\Lambda\varepsilon$, which follows
from (12), and $H_{\max}(\psi_{\varepsilon}^{\nu})\leq R^{2}$. For the second,
we instead use that
$H_{\max}(\psi_{\varepsilon}^{\nu})\leq\varepsilon/\lambda$. ∎

###  3.1 Proof of Theorem 3.1

Our proof relies on three propositions. To continue, we require the following
objects. Let $\tau\in\Gamma(\mu,\nu)$ be a fixed (though not necessarily
unique) optimal transport coupling between $\mu$ and $\nu$. For
$z\in\mathbb{R}^{d}$, let $\tau(\cdot|z)$ be associated (regular) conditional
measure (see Bogachev, 2007, Chapter 10), so that for all measurable
$f:\mathbb{R}^{d}\times\mathbb{R}^{d}\to[0,+\infty)$

$\iint f(y,z)\,\mathrm{d}\tau(y,z)=\int\left(\int
f(y,z)\,\mathrm{d}\tau(y|z)\right)\,\mathrm{d}\nu(z)\,.$

For $x\in\mathbb{R}^{d}$, let $Q(\cdot|x)$ be the probability measure with

$\displaystyle\forall z\in\mathbb{R}^{d},\
\frac{\,\mathrm{d}Q(\cdot|x)}{\,\mathrm{d}\nu}(z)\coloneqq\int\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y|z)\,,$
(13)

where $\gamma_{\varepsilon}^{\mu}(x,y)$ is the density of
$\pi_{\varepsilon}^{\mu}$ w.r.t. $\rho\otimes\mu$. Note that this indeed
defines a density as we have the relation

$\int\left(\int\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y|z)\right)\,\mathrm{d}\nu(z)=\iint\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)=\int\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\mu(y)=1\,.$

We also define the conditional Kullback–Leibler divergence:

$\displaystyle
I\coloneqq\int\text{KL}(Q(\cdot|x)\|\pi_{\varepsilon}^{\nu}(\cdot|x))\,\mathrm{d}\rho(x)\,,$
(14)

We are now in a position to proceed with the proof. First, we decompose the
difference of forward entropic Brenier maps into a $W_{2}(\mu,\nu)$ term, plus
a term depending on $I$.

######  Proposition 3.6.

Suppose $\rho,\mu,\nu$ have finite second moment. Then

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq
W_{2}(\mu,\nu)+(2H_{\max}(\varphi_{\varepsilon}^{\nu})I)^{1/2}\,.$

###### Proof of 3.6.

We assume that $\varphi_{\varepsilon}^{\nu}$ is finite everywhere, for
otherwise there is nothing to prove. We have that

$\displaystyle T_{\varepsilon}^{\mu}(x)-T_{\varepsilon}^{\nu}(x)=\int
y\,\mathrm{d}\pi_{\varepsilon}^{\mu}(y|x)-\int
z\,\mathrm{d}\pi_{\varepsilon}^{\nu}(z|x)$ $\displaystyle=\iint
y\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)-\int
z\,\mathrm{d}\pi_{\varepsilon}^{\nu}(z|x)$
$\displaystyle=\iint(y-z)\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)+\iint
z(\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)-\,\mathrm{d}\pi_{\varepsilon}^{\nu}(z|x))$
$\displaystyle=\iint(y-z)\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)+\int
z\,\mathrm{d}(Q(z|x)-\pi_{\varepsilon}^{\nu}(z|x))\,.$

Taking the $L^{2}(\rho)$-norm of both sides and applying Minkowski’s and
Jensen’s inequalities yields

$\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\leq\left(\iint\|y-z\|^{2}\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)\,\mathrm{d}\rho(x)\right)^{1/2}\\\
+\Bigl{\|}\int
z\,\mathrm{d}(Q(z|\cdot)-\pi_{\varepsilon}^{\nu}(z|\cdot))\Bigr{\|}_{L^{2}(\rho)}\,.$

Since $\tau$ is an optimal coupling between $\mu$ and $\nu$ and
$\int\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\rho(x)=1$, the first term is
$W_{2}(\mu,\nu)$. For the second term, Corollary 2.3 implies for all
$x\in\mathbb{R}^{d}$

$\Bigl{\|}\int
z\,\mathrm{d}(Q(z|x)-\pi_{\varepsilon}^{\nu}(z|x))\Bigr{\|}\leq\sqrt{2H_{\max}(\varphi_{\varepsilon}^{\nu})\text{KL}(Q(\cdot|x)\|\pi_{\varepsilon}^{\nu}(\cdot|x))}\,.$

Therefore

$\displaystyle\Bigl{\|}\int
z\,\mathrm{d}(Q(z|\cdot)-\pi_{\varepsilon}^{\nu}(z|\cdot))\Bigr{\|}_{L^{2}(\rho)}$
$\displaystyle\leq\sqrt{2H_{\max}(\varphi_{\varepsilon}^{\nu})\int\text{KL}(Q(\cdot|x)\|\pi_{\varepsilon}^{\nu}(\cdot|x))\,\mathrm{d}\rho(x)}$
$\displaystyle=(2H_{\max}(\varphi_{\varepsilon}^{\nu})I)^{1/2}\,,$

which completes the proof. ∎

In (14), we defined the conditional relative entropy between $Q(\cdot|x)$ and
the conditional entropic coupling $\pi_{\varepsilon}^{\nu}(\cdot|x)$. We now
turn to directly bounding the quantity $I$. Note that for all
$z\in\mathbb{R}^{d}$

$\displaystyle\frac{\,\mathrm{d}Q(\cdot|x)}{\,\mathrm{d}\pi_{\varepsilon}^{\nu}(\cdot|x)}(z)=\frac{\int\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y|z)}{\gamma_{\varepsilon}^{\nu}(x,z)}\,.$

An application of Jensen’s inequality then yields that

$\displaystyle
I\leq\bar{I}\coloneqq\iiint\log\Bigl{(}\frac{\gamma_{\varepsilon}^{\mu}(x,y)}{\gamma_{\varepsilon}^{\nu}(x,z)}\Bigr{)}\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)\,\mathrm{d}\rho(x)\,.$
(15)

Expanding the densities $\gamma_{\varepsilon}^{\mu}(x,y)$ and
$\gamma_{\varepsilon}^{\nu}(x,z)$ and performing the integration, we obtain

$\displaystyle\varepsilon\bar{I}$ $\displaystyle=\iiint\langle
x,y-z\rangle\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\tau(y,z)\,\mathrm{d}\rho(x)+\int\varphi_{\varepsilon}^{\nu}\,\mathrm{d}\rho+\int\psi_{\varepsilon}^{\nu}\,\mathrm{d}\nu-\int\varphi_{\varepsilon}^{\mu}\,\mathrm{d}\rho-\int\psi_{\varepsilon}^{\mu}\,\mathrm{d}\mu$
$\displaystyle=\iint\langle
S_{\varepsilon}^{\mu}(y),y-z\rangle\,\mathrm{d}\tau(y,z)+\int\varphi_{\varepsilon}^{\nu}\,\mathrm{d}\rho+\int\psi_{\varepsilon}^{\nu}\,\mathrm{d}\nu-\int\varphi_{\varepsilon}^{\mu}\,\mathrm{d}\rho-\int\psi_{\varepsilon}^{\mu}\,\mathrm{d}\mu$

where we use the equality $S_{\varepsilon}^{\mu}(y)=\int
x\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\rho(x)$ in the last line. If we
define $\tilde{I}$ as a symmetric analogue to $\bar{I}$, namely,

$\displaystyle\tilde{I}\coloneqq\iiint\log\Bigl{(}\frac{\gamma_{\varepsilon}^{\nu}(x,z)}{\gamma_{\varepsilon}^{\mu}(x,y)}\Bigr{)}\gamma_{\varepsilon}^{\nu}(x,z)\,\mathrm{d}\tau(y,z)\,\mathrm{d}\rho(x)\,,$

then, since $0\leq\tilde{I}$, a symmetric calculation immediately yields the
following:

######  Proposition 3.7.

Suppose $\rho,\mu,\nu$ have finite second moment. Then

$\displaystyle\varepsilon\bar{I}\leq\varepsilon(\bar{I}+\tilde{I})=\iint\langle
S_{\varepsilon}^{\mu}(y)-S_{\varepsilon}^{\nu}(z),y-z\rangle\,\mathrm{d}\tau(y,z)\,.$
(16)

All in all, our bound currently reads

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}$
$\displaystyle\leq W_{2}(\mu,\nu)$
$\displaystyle\qquad+(2\varepsilon^{-1}H_{\max}(\varphi_{\varepsilon}^{\nu}))^{1/2}\Bigl{(}\iint\langle
S_{\varepsilon}^{\mu}(y)-S_{\varepsilon}^{\nu}(z),y-z\rangle\,\mathrm{d}\tau(y,z)\Bigr{)}^{1/2}\,.$

The second term depends on the two backward entropic Brenier maps,
$S_{\varepsilon}^{\mu}$ and $S_{\varepsilon}^{\nu}$. The next proposition
shows that tilt stability can again be used to bound the difference between
these backward maps by $\bar{I}$.

######  Proposition 3.8.

Suppose $\rho,\mu,\nu$ have finite second moment. Then

$\displaystyle\iint\|S_{\varepsilon}^{\mu}(y)-S_{\varepsilon}^{\nu}(z)\|^{2}\,\mathrm{d}\tau(y,z)\leq
2H_{\max}(\psi_{\varepsilon}^{\nu})\bar{I}\,.$

###### Proof.

We assume that $\psi_{\varepsilon}^{\nu}$ is finite everywhere, for otherwise
there is nothing to prove. Recall that $S_{\varepsilon}^{\nu}(z)=\int
x\,\mathrm{d}\pi_{\varepsilon}^{\nu}(x|z)$ and similarly for
$S_{\varepsilon}^{\mu}(y)$. By Corollary 2.3, we have the following bound

$\displaystyle\|S_{\varepsilon}^{\nu}(z)-S_{\varepsilon}^{\mu}(y)\|^{2}$
$\displaystyle=\Bigl{\|}\int
x\,\mathrm{d}(\pi_{\varepsilon}^{\nu}(x|z)-\pi_{\varepsilon}^{\mu}(x|y))\Bigr{\|}^{2}$
$\displaystyle\leq
2H_{\max}(\psi_{\varepsilon}^{\nu})\text{KL}(\pi_{\varepsilon}^{\mu}(\cdot|y)\|\pi_{\varepsilon}^{\nu}(\cdot|z))$
$\displaystyle=2H_{\max}(\psi_{\varepsilon}^{\nu})\int\log\Bigl{(}\frac{\gamma_{\varepsilon}^{\mu}(x,y)}{\gamma_{\varepsilon}^{\nu}(x,z)}\Bigr{)}\gamma_{\varepsilon}^{\mu}(x,y)\,\mathrm{d}\rho(x)\,.$

Integrating with respect to $\tau$ concludes the proof. ∎

Finally, with these results in hand, we can prove our main result.

###### Proof of Theorem 3.1.

From (16), we apply Cauchy-Schwarz, resulting in

$\displaystyle\varepsilon\bar{I}$ $\displaystyle\leq\iint\langle
S_{\varepsilon}^{\mu}(y)-S_{\varepsilon}^{\nu}(z),y-z\rangle\,\mathrm{d}\tau(y,z)\leq
W_{2}(\mu,\nu)(2H_{\max}(\psi_{\varepsilon}^{\nu})\bar{I})^{1/2}\,.$

This ultimately implies

$\displaystyle
I^{1/2}\leq\bar{I}^{1/2}\leq\varepsilon^{-1}W_{2}(\mu,\nu)(2H_{\max}(\psi_{\varepsilon}^{\nu}))^{1/2}\,,$

where we recall the first inequality from (15). Together with 3.6, the proof
is complete. ∎

##  4 Application: Improved quantitative stability of semi-discrete optimal
transport maps

As an application of our new stability results for entropic Brenier maps, we
turn to proving quantitative stability results for _unregularized_ optimal
transport maps. As highlighted in the introduction, our proof technique for
proving quantitative stability for optimal transport maps is based on the
following decomposition:

$\displaystyle\begin{split}\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}&\leq\|T_{0}^{\mu}-T_{\varepsilon}^{\mu}\|_{L^{2}(\rho)}+\|T_{0}^{\nu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\\\
&\qquad+\|T_{\varepsilon}^{\mu}-T_{\varepsilon}^{\nu}\|_{L^{2}(\rho)}\,.\end{split}$
(17)

Recall that Corollary 3.3 takes care of the last term under virtually no
assumptions other than boundedness of the measures. It remains to control the
first two terms in the above decomposition, also known as _bias_ terms.

To the best of our knowledge, bounds on the bias of entropic Brenier maps are
known only under strong assumptions. For example, Pooladian and Niles-Weed
(2021, Corollary 1) show that

$\displaystyle\|T_{0}^{\mu}-T_{\varepsilon}^{\mu}\|_{L^{2}(\rho)}^{2}\lesssim\varepsilon^{2}I_{0}(\rho,\mu)\,,$

where $I_{0}(\rho,\mu)$ is the integrated Fisher information along the
Wasserstein geodesic between $\rho$ and $\mu$, where $\rho$ and $\mu$ have
upper and lower bounded densities over compact domains. Such assumptions,
while essential for estimating optimal transport maps on the basis of samples,
are too restrictive for our purposes.‡‡‡Indeed, under these assumptions, it is
well-known via Caffarelli regularity theory (Caffarelli, 1992, 1996) that the
corresponding optimal transport map is Lipschitz, so the results of Gigli
(2011) already imply a stability bound.

With this in mind, our goal is to establish quantitative control on the bias
of entropic Brenier maps under less restrictive regularity conditions.
Specifically, we turn to the _semi-discrete_ setting, where $\rho$ has a
density, and $\mu$ and $\nu$ are both discrete measures. As we will shortly
see, this setting allows for (17) to be used to obtain meaningful bounds on
the stability of two semi-discrete optimal transport maps when the discrete
measures themselves have favorable properties.

We briefly recall some fundamental notions from semi-discrete optimal
transport: let $\mu=\sum_{j=1}^{J}\mu_{j}\delta_{y_{j}}$ be a discrete
probability measure with atoms located at the points $\\{y_{j}\\}_{j=1}^{J}$
with corresponding weights $\mu_{j}>0$. In this setting, the Brenier potential
is given explicitly by

$\displaystyle\varphi_{0}^{\mu}(x)=\max_{j\in\\{1,\ldots,J\\}}\langle
x,y_{j}\rangle-(\psi_{0}^{\mu})_{j}\,,$

where $\psi_{0}^{\mu}\in\mathbb{R}^{J}$ is the dual potential. Note that
$\varphi_{0}^{\mu}$ is $\rho$-almost everywhere differentiable, and so the
Brenier map $T_{0}^{\mu}=\nabla\varphi_{0}^{\mu}$ is well-defined. The inverse
transport map is now set-valued, where for a given target atom $y_{j}$, we
define the _Laguerre cell_ $L_{j}\coloneqq(T_{0}^{\mu})^{-1}(y_{j})$. These
cells partition the support of $\rho$. Consequently, for $x\in L_{j}$, the
optimal transport mapping is $x\mapsto T_{0}^{\mu}(x)=y_{j}$.

With these notions in hand, we are ready to present the following result on
the convergence of entropic Brenier maps to their unregularized counterpart;
its proof is located in Appendix A. This is a slightly different version of
Pooladian et al. (2023, Theorem 3.4), based off the results of Delalande
(2022).

######  Proposition 4.1 (Quantitative bias in the semi-discrete setting).

Let $\rho$ be a compactly supported probability distribution with a density
over $\mathbb{R}^{d}$ and $\mu$ be a discrete measure, written
$\mu=\sum_{j=1}^{J}\mu_{j}\delta_{y_{j}}$. Then for all $\varepsilon>0$,

$\displaystyle\|T_{0}^{\mu}-T_{\varepsilon}^{\mu}\|_{L^{2}(\rho)}^{2}\leq
e^{2\|\psi_{0}^{\mu}-\psi_{\varepsilon}^{\mu}\|_{\infty}/\varepsilon}\varepsilon\sum_{i,j}\frac{\|y_{i}-y_{j}\|}{2}\int_{0}^{\infty}\\!\\!\\!h_{ij}(u\varepsilon)\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{u/2}\Bigr{)}^{-1}\,\mathrm{d}u\,,$
(18)

where $0\leq h_{ij}(\cdot)$ measures the amount of overlap between $L_{i}$ and
$L_{j}$ weighted against the source measure $\rho$ (see (21) in the appendix
for precise details).

In addition, suppose that

(S1) the density of $\rho$ has convex support in $B(0;R)$, is $\alpha$-Hölder
continuous for $\alpha\in(0,1]$, and there exist $\rho_{\min},\rho_{\max}$
such that $0<\rho_{\min}\leq\rho(x)\leq\rho_{\max}$,

(S2) the support of $\mu$ lies in $B(0;R)$, and all the weights are uniformly
lower-bounded i.e., $\mu_{j}\geq\mu_{\min}>0$ for all $j\in\\{1,\ldots,J\\}$.

Then it holds that, for all $\varepsilon>0$

$\displaystyle\|T_{0}^{\mu}-T_{\varepsilon}^{\mu}\|^{2}_{L^{2}(\rho)}\leq
C_{0}e^{C_{1}\varepsilon^{\alpha}}\varepsilon\,,$ (19)

where the constants depend on $\rho_{\min},\rho_{\max},d,R,\mu_{\min},J$,
$\min_{i\neq j}\|y_{i}-y_{j}\|$, and on the maximum angle formed by three non
aligned points among the atoms $\\{y_{j}\\}_{j=1}^{J}$.

######  Remark 4.2.

Following the asymptotic results of Altschuler et al. (2022), we can take a
limit of (18), resulting in the computation

$\displaystyle\limsup_{\varepsilon\to\infty}\varepsilon^{-1}\|T_{0}^{\mu}-T_{\varepsilon}^{\mu}\|^{2}_{L^{2}(\rho)}$
$\displaystyle=\sum_{i,j}\frac{\|y_{i}-y_{j}\|h_{ij}(0)}{2}\int_{0}^{\infty}\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{u/2}\Bigr{)}^{-1}\,\mathrm{d}u$
$\displaystyle=\sum_{i,j}\|y_{i}-y_{j}\|h_{ij}(0)\log(1+\mu_{j}/\mu_{i})\,,$

where we used that $t\mapsto h_{ij}(t)$ is continuous at $t=0$ (which holds
if, for instance, $\rho$ has an upper bounded density with compact support).
We conjecture that this quantity is uniformly bounded for all discrete
measures.

Combined with (17) and Corollary 3.3, we can state and prove our main theorem
for this section. While the conditions do not allow for arbitrary discrete
measures, we stress that they permit a wide class of discrete measures, and in
particular measures supported on different masses. To our knowledge, this is
the first general improvement to the stability bound of Delalande and Mérigot
(2023), even in the semi-discrete case.

######  Theorem 4.3 (Near-tight stability in the semi-discrete setting).

Suppose $\rho$ satisfies (S1), and both $\mu$ and $\nu$ each independently
satisfy (S2) (with possibly all different parameters). Then

$\displaystyle\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}\lesssim
W_{2}^{1/3}(\mu,\nu)\,,$

where the underlying constants depend on those from 4.1.

###### Proof.

First, we note that if $W_{2}(\mu,\nu)\geq 1$, then
$\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}\leq 2R\leq 2RW_{2}(\mu,\nu)^{1/3}$,
so the only case of interest is when $W_{2}(\mu,\nu)\leq 1$. Then, using the
decomposition (17) with Corollary 3.3 and two applications of (19), we obtain

$\displaystyle\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}^{2}$
$\displaystyle\leq
4\max\\{C_{0}(\mu)e^{C_{1}(\mu)\varepsilon^{\alpha}},C_{0}(\nu)e^{C_{1}(\nu)\varepsilon^{\alpha}}\\}\varepsilon+(2+4R^{4}/\varepsilon^{2})W_{2}^{2}(\mu,\nu)\,.$

Choosing $\varepsilon=W_{2}^{2/3}(\mu,\nu)\leq 1$, we obtain our desired rate
with a prefactor scaling like $C_{0}e^{C_{1}}+1$, where we choose the worse
constant arising from the bias terms between $\mu$ and $\nu$. ∎

######  Remark 4.4.

Closest to this result is that of Bansil and Kitagawa (2022): when $\mu$ and
$\nu$ are supported on the _same_ atoms, the following bound holds§§§This
bound is not explicitly written in the paper but it can be extracted from
their Theorem 1.3.

$\displaystyle\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}^{2}\leq(J-1)\text{diam}(\Omega)^{2}\mathrm{d_{TV}}(\mu,\nu)\,,$
(20)

where $\mathrm{d_{TV}}(\cdot,\cdot)$ is the total variation distance, and $J$
is the number of atoms in the support of $\mu$ and $\nu$. Theorem 4.3 implies
meaningful bounds in some situations in which (20) fails to do so. For
example, consider the simple setting where $\rho=\text{Unif}(B(0,1))$ and
$\mu_{\theta}=\tfrac{1}{2}(\delta_{e_{\theta}}+\delta_{-e_{\theta}})$ with
$e_{\theta}=(\cos(\theta),\sin(\theta))$ for $0<\theta\ll\pi/2$. It is easy to
verify that $W_{2}(\mu_{0},\mu_{\theta})\asymp\theta$, so Theorem 4.3 gives
$\|T_{0}^{\mu}-T_{0}^{\nu}\|_{L^{2}(\rho)}\lesssim\theta^{1/3}$. On the other
hand, since $\mathrm{d_{TV}}(\mu_{0},\mu_{\theta})=1$, the bound (20) is
vacuous.

##  5 Conclusion

In this work, we provide a tight characterization of the stability of entropic
Brenier maps under variations of the target measure. Our bounds do not require
any of the measures to have densities, and are strengthened when the entropic
Brenier potentials are smooth. As an application, we used our entropic bounds
combined with existing bounds on the bias of semi-discrete entropic Brenier
maps to prove improved stability estimates for semi-discrete transport maps
when the discrete measures and smooth source measure satisfy certain
regularity assumptions.

## Acknowledgements

AAP thanks Sinho Chewi for helpful discussions. JNW is supported by the Sloan
Research Fellowship and NSF grant DMS-2210583. AAP thanks NSF grant
DMS-1922658 and Meta AI Research for financial support.

##  Appendix A Proof of the bias term

Recall that our target measures are discrete measures of the form

$\displaystyle\mu=\sum_{j=1}^{J}\mu_{j}\delta_{y_{j}}\,.$

and that we write the Laguerre cells as $L_{i}$ for $i\in\\{1,\dots,J\\}$.

We require the following definitions, which we borrow from Altschuler et al.
(2022). For $x\in L_{i}$ and any other $j\in\\{1,\dots,J\\}$, we write

$\displaystyle\Delta_{ij}(x)\coloneqq 2(\langle
x,y_{i}-y_{j}\rangle-\psi_{0}^{\mu}(y_{i})+\psi_{0}^{\mu}(y_{j}))\,,$

and $H_{ij}(t)=\\{x\in L_{i}:\ \Delta_{ij}(x)=t\\}$, which represents the
trace on $L_{i}$ of the hyperplane spanned by the boundary between $L_{i}$ and
$L_{j}$, shifted by $t$ (should the two cells have non-empty intersection).
Moreover, we have the following co-area formula: for every nonnegative
measurable function $f:\mathbb{R}\to{\mathbb{R}_{+}}$,

$\int_{L_{i}}f(\Delta_{ij}(x))\rho(x)\,\mathrm{d}x=\frac{1}{2\|y_{i}-y_{j}\|}\int_{0}^{\infty}f(t)h_{ij}(t)\,\mathrm{d}t,$

where

$\displaystyle
h_{ij}(t)=\int_{H_{ij}(t)}\rho(x)\,\mathrm{d}\mathcal{H}_{d-1}(x)\,,$ (21)

and $\mathcal{H}_{d-1}$ is the $(d-1)$-dimensional Hausdorff measure.

###### Proof of 4.1.

Let $i\in\\{1,\ldots,J\\}$ and let $x\in L_{i}$. For $j\in\\{1,\dots,J\\}$
other than $i$, we have the upper bound

$\displaystyle\pi_{\varepsilon}^{\mu}(y_{j}|x)$
$\displaystyle=\mu_{i}e^{(\langle
x,y_{j}\rangle-\varphi_{\varepsilon}^{\mu}(x)-\psi_{\varepsilon}^{\mu}(y_{j}))/\varepsilon}$
$\displaystyle=\frac{\mu_{i}e^{(\langle
x,y_{j}\rangle-\psi_{\varepsilon}^{\mu}(y_{j}))/\varepsilon}}{\sum_{k=1}^{J}\mu_{k}e^{(\langle
x,y_{k}\rangle-\psi_{\varepsilon}^{\mu}(y_{k}))/\varepsilon}}$
$\displaystyle\leq\frac{\mu_{j}e^{(\langle
x,y_{j}\rangle-\psi_{\varepsilon}^{\mu}(y_{j}))/\varepsilon}}{\mu_{i}e^{(\langle
x,y_{i}\rangle-\psi_{\varepsilon}^{\mu}(y_{i}))/\varepsilon}+\mu_{j}e^{(\langle
x,y_{j}\rangle-\psi_{\varepsilon}^{\mu}(y_{j}))/\varepsilon}}\,.$

Adding and subtracting appropriate factors of $\psi_{0}^{\mu}(y_{i})$ and
$\psi_{0}^{\mu}(y_{j})$, we obtain

$\displaystyle\pi_{\varepsilon}^{\mu}(y_{j}|x)$ $\displaystyle\leq
e^{2\|\psi_{0}^{\mu}-\psi_{\varepsilon}^{\mu}\|_{\infty}/\varepsilon}\frac{\mu_{j}e^{(\langle
x,y_{j}\rangle-\psi_{0}^{\mu}(y_{j}))/\varepsilon}}{\mu_{i}e^{(\langle
x,y_{i}\rangle-\psi_{0}^{\mu}(y_{i}))/\varepsilon}+\mu_{j}e^{(\langle
x,y_{j}\rangle-\psi_{0}^{\mu}(y_{j}))/\varepsilon}}$
$\displaystyle=e^{2\|\psi^{\mu}_{0}-\psi^{\mu}_{\varepsilon}\|_{\infty}/\varepsilon}\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{\Delta_{ij}(x)/2\varepsilon}\Bigr{)}^{-1}\,,$

By an application of Jensen’s inequality, we have

$\displaystyle\|T_{\varepsilon}^{\mu}(x)-y_{i}\|^{2}$
$\displaystyle\leq\sum_{j=1}^{J}\pi_{\varepsilon}^{\mu}(y_{j}|x)\|y_{i}-y_{j}\|^{2}$
$\displaystyle\leq
e^{2\|\psi^{\mu}_{0}-\psi^{\mu}_{\varepsilon}\|_{\infty}/\varepsilon}\sum_{j=1}^{J}\|y_{i}-y_{j}\|^{2}\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{\Delta_{ij}(x)/2\varepsilon}\Bigr{)}^{-1}\,,$

so integrating against $\rho$ (partitioned into the $J$ Laguerre cells) yields

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{0}^{\mu}\|^{2}_{L^{2}(\rho)}$
$\displaystyle\leq
e^{2\|\psi^{\mu}_{0}-\psi^{\mu}_{\varepsilon}\|_{\infty}/\varepsilon}\sum_{i,j}\|y_{i}-y_{j}\|^{2}\int_{L_{i}}\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{\Delta_{ij}(x)/2\varepsilon}\Bigr{)}^{-1}\,\mathrm{d}\rho(x)$
$\displaystyle=e^{2\|\psi^{\mu}_{0}-\psi^{\mu}_{\varepsilon}\|_{\infty}/\varepsilon}\sum_{i,j}\|y_{i}-y_{j}\|/2\int_{0}^{\infty}h_{ij}(t)\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{t/2\varepsilon}\Bigr{)}^{-1}\,\mathrm{d}t$
$\displaystyle=e^{2\|\psi^{\mu}_{0}-\psi^{\mu}_{\varepsilon}\|_{\infty}/\varepsilon}\varepsilon\sum_{i,j}\|y_{i}-y_{j}\|/2\int_{0}^{\infty}h_{ij}(u\varepsilon)\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{u/2}\Bigr{)}^{-1}\,\mathrm{d}u\,,$

where the second line follows from the definition of the co-area formula, and
the last line is a change of variables $u=t/\varepsilon$. This gives (18).

With the additional assumptions (S1) and (S2), we can use Corollary 3.2 of
Delalande (2022), which tells us that

$\displaystyle\varepsilon^{-1}\|\psi_{0}^{\mu}-\psi_{\varepsilon}^{\mu}\|_{\infty}\leq
C_{1}\varepsilon^{\alpha}\,,$ (22)

where the underlying constant depends on $d,R,J,\mu_{\min},\min_{i\neq
j}\|y_{i}-y_{j}\|,\rho_{\min},\rho_{\max}$, and on the maximum angle formed by
three non aligned points among the atoms $\\{y_{j}\\}_{j=1}^{J}$. This gives
and upper bound of

$\displaystyle\|T_{\varepsilon}^{\mu}-T_{0}^{\mu}\|^{2}_{L^{2}(\rho)}\leq
e^{C_{1}\varepsilon^{\alpha}}\Bigl{(}\sum_{i,j}\|y_{i}-y_{j}\|/2\int_{0}^{\infty}h_{ij}(u\varepsilon)\Bigl{(}1+\tfrac{\mu_{i}}{\mu_{j}}e^{u/2}\Bigr{)}^{-1}\,\mathrm{d}u)\varepsilon\,.$

Since $\|y_{i}-y_{j}\|\leq 2R$, the masses $\mu_{i},\mu_{j}$ are larger than
$\mu_{\min}$, and $h_{ij}(u\varepsilon)$ is bounded under our assumptions on
$\rho$, the proof is concluded. ∎

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