Quantitative stability of barycenters in the Wasserstein space

Abstract

Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a Hölder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that allow to quantify the strong convexity of the barycenter functional. Consequences regarding the statistical estimation of Wasserstein barycenters and the convergence of regularized Wasserstein barycenters towards their non-regularized counterparts are explored.

Appendices

Appendix A. Dual formulation for the Wasserstein barycenter problem

Proof of Proposition 1.1

Instead of showing directly the formulation of Proposition 1.1, we will rather show

$$\begin{aligned} \min _{\mu \in \mathcal {P}(\Omega )} F_\mathbb {P}(\mu )&= \max \Bigg \{ \int _{\mathcal {P}(\Omega )} \langle \phi ^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ) \mid (\phi _\rho )_\rho \in \textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega )), \\ {}&\quad \int _{\mathcal {P}(\Omega )} \phi _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) = 0 \Bigg \}, \end{aligned}$$

where for any $\rho \in \mathcal {P}(\Omega )$, $\phi _\rho ^c$ denotes the following c-transform of $\phi _\rho$: $\phi _\rho ^c(\cdot ) = \inf _{y \in \Omega } \frac{1}{2} \left\| \cdot -y\right\| ^2 - \phi _\rho (y)$. Such a formulation entails the result of Proposition 1.1 by the change of variable $(\psi _\rho )_\rho = \frac{\left\| \cdot \right\| ^2}{2} - (\phi _\rho )_\rho \in \textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$.

Duality Let’s first show that the value of $\min _{\mu \in \mathcal {P}(\Omega )} F_\mathbb {P}(\mu )$ is equal to the value of the following supremum

$$\begin{aligned} \mathrm {(D)_\mathbb {P}}' := \sup \Bigg \{&\int _{\mathcal {P}(\Omega )} \langle \phi ^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ) \mid (\phi _\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega )), \quad \int _{\mathcal {P}(\Omega )} \phi _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) = 0 \Bigg \}, \end{aligned}$$

where $\textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega ))$ denotes the set of $\mathbb {P}$-measurable and Bochner integrable mappings from $\mathcal {P}(\Omega )$ to the space $(\mathcal {C}(\Omega ), \left\| \cdot \right\| _\infty )$ of continuous function from $\Omega$ to $\mathbb {R}$ equipped with the supremum norm. Introduce the functional $H: \mathcal {C}(\Omega ) \rightarrow \mathbb {R}$ defined for all $\varphi \in \mathcal {C}(\Omega )$ by

$$\begin{aligned} H(\varphi ) = \inf \Bigg \{&-\int _{\mathcal {P}(\Omega )} \langle \phi ^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ) \mid (\phi _\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega )), \quad \int _{\mathcal {P}(\Omega )} \phi _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) = \varphi (\cdot ) \Bigg \}. \end{aligned}$$

Notice then that $\mathrm {(D)_\mathbb {P}}' = -H(0)$. On the other hand, notice that H has the following convex conjugate: for $\mu \in \mathcal {P}(\Omega )$,

$$\begin{aligned} H^*(\mu )&= \sup \left\{ \langle \varphi | \mu \rangle - H(\varphi ) \mid \varphi \in \mathcal {C}(\Omega ) \right\} \\&= \sup \Bigg \{ \langle \varphi | \mu \rangle + \int _{\mathcal {P}(\Omega )} \langle \phi ^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ) \mid \varphi \in \mathcal {C}(\Omega ), (\phi _\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega )),\\&\quad \int _{\mathcal {P}(\Omega )} \phi _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) = \varphi (\cdot ) \Bigg \} \\&= \sup \left\{ \int _{\mathcal {P}(\Omega )} \langle \phi _\rho | \mu \rangle \textrm{d}\mathbb {P}(\rho ) + \int _{\mathcal {P}(\Omega )} \langle \phi ^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ), \quad (\phi _\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega )) \right\} \\&= \int _{\mathcal {P}(\Omega )} \left( \sup _{\phi _\rho \in \mathcal {C}(\Omega )} \langle \phi _\rho | \mu \rangle + \langle \phi ^c_\rho | \rho \rangle \right) \textrm{d}\mathbb {P}(\rho ) \\&= \int _{\mathcal {P}(\Omega )} \frac{1}{2} \textrm{W}_2^2(\mu , \rho ) \textrm{d}\mathbb {P}(\rho ), \end{aligned}$$

where we used the Kantorovich duality formula (see for instance [43]) to get to the last line. We thus have

$$\begin{aligned} \min _{\mu \in \mathcal {P}(\Omega )} F_\mathbb {P}(\mu ) = \inf _{\mu \in \mathcal {P}(\Omega )} H^*(\mu ) = - H^{**}(0). \end{aligned}$$

Therefore, showing that $\mathrm {(D)_\mathbb {P}}' = \min _{\mu \in \mathcal {P}(\Omega )} F_\mathbb {P}(\mu )$ corresponds to show that $H(0) = H^{**}(0)$. Since H is convex (by concavity of the c-transform operation), this will follow from the continuity of H at 0 for the supremum-norm over $\mathcal {C}(\Omega )$ (Proposition 4.1 of [21]). For this, we can first notice that H never takes the value $-\infty$: for any $\varphi \in \mathcal {C}(\Omega )$ and $(\phi _\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega ))$ such that $\int _{\mathcal {P}(\Omega )} \phi _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) = \varphi (\cdot )$, one has

$$\begin{aligned} \forall \rho \in \mathcal {P}(\Omega ),\quad -\phi _\rho ^c(x) = \sup _{y \in \mathbb {R}^d} \phi _\rho (y) - \frac{1}{2}\left\| x - y\right\| ^2 \ge \phi _\rho (0) - \frac{1}{2} \left\| x\right\| ^2. \end{aligned}$$

If follows that

$$\begin{aligned} H(\varphi ) \ge \varphi (0) - \int _{\mathcal {P}(\Omega )} \frac{M_2(\rho )}{2} \textrm{d}\mathbb {P}(\rho ) > -\infty . \end{aligned}$$

On the other hand, notice that H is bounded from above in a neighborhood of 0 in $\mathcal {C}(\Omega )$: for any $\varphi \in \mathcal {C}(\Omega )$ such that $\left\| \varphi \right\| _\infty \le 1$, one has $-\varphi ^c(x) \le 1$ for any $x \in \mathbb {R}^d$ so that

$$\begin{aligned} H(\varphi ) \le - \int _{\mathcal {P}(\Omega )} \langle (\varphi )^c | \rho \rangle \textrm{d}\mathbb {P}(\rho ) \le 1. \end{aligned}$$

A standard convex analysis result (Proposition 2.5 in [21]) then ensures that H is continuous at 0, so that $H(0) = H^{**}(0)$ and $\mathrm {(D)_\mathbb {P}}' = \min _{\mu \in \mathcal {P}(\Omega )} F_\mathbb {P}(\mu )$.

Restriction to $\textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$. We show here that we can run the supremum $\mathrm {(D)_\mathbb {P}}'$ only over $\textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$ instead of $\textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega ))$, that is

$$\begin{aligned} \mathrm {(D)_\mathbb {P}}'&= \sup \Bigg \{ \int _{\mathcal {P}(\Omega )} \langle \phi ^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ) \mid (\phi _\rho )_\rho \in \textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega )),\\ {}&\qquad \times \int _{\mathcal {P}(\Omega )} \phi _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) = 0 \Bigg \}. \end{aligned}$$

Let $(\phi _\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega ))$ be an admissible solution to $\mathrm {(D)_\mathbb {P}}'$, i.e. $(\phi _\rho )_\rho$ satisfies

$$\begin{aligned} \int _{\mathcal {P}(\Omega )} \phi _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) = 0. \end{aligned}$$

(13)

Then we can build from $(\phi _\rho )_\rho$ another admissible solution $(\tilde{\phi }_\rho )_\rho$ that belongs to $\textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$ and that performs better at $\mathrm {(D)_\mathbb {P}}'$, i.e. that verifies

$$\begin{aligned} \int _{\mathcal {P}(\Omega )} \langle \tilde{\phi }^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ) \ge \int _{\mathcal {P}(\Omega )} \langle \phi ^c_\rho | \rho \rangle \textrm{d}\mathbb {P}(\rho ). \end{aligned}$$

(14)

Indeed, introduce $(\hat{\phi }_\rho )_\rho := (\phi ^{cc}_\rho )_\rho$. Then for all $\rho \in \mathcal {P}(\Omega )$, $\hat{\phi }_\rho = \phi ^{cc}_\rho$ is obviously 2R-Lipschitz (as a c-transform) and satisfies $\hat{\phi }_\rho ^c = \phi _\rho ^c$ and $\hat{\phi }_\rho \ge \phi _\rho$ (as a double c-transform). Using then (13), one has that

$$\begin{aligned} \alpha (\cdot ):= \int _{\mathcal {P}(\Omega )} \hat{\phi }_\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) \ge 0, \end{aligned}$$

where $\alpha$ is also 2R-Lipschitz. Now denoting $\tilde{\phi }_\rho = \hat{\phi _\rho } - \alpha$ for all $\rho \in \mathcal {P}(\Omega )$, the mapping $(\tilde{\phi }_\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega ))$ is admissible to $\mathrm {(D)_\mathbb {P}}'$ by construction and satisfies $\tilde{\phi }_\rho \le \hat{\phi }_\rho$ for all $\rho \in \mathcal {P}(\Omega )$, so that $\tilde{\phi }^c_\rho \ge \hat{\phi }^c_\rho = \phi ^c_\rho$ (using that the c-transform is order-reversing). For each $\rho \in \mathcal {P}(\Omega )$, up to subtracting $\tilde{\phi }_\rho (0)$ to $\tilde{\phi }_\rho$ (this operation leaves $(\tilde{\phi }_\rho )_\rho$ admissible to $\mathrm {(D)_\mathbb {P}}'$ and does not change its value), one can assume that $\tilde{\phi }_\rho (0) = 0$. Noticing that $\tilde{\phi }_\rho$ is 4R-Lipschitz by construction, we have the bound $\left\| \tilde{\phi }_\rho \right\| _{W^{1, \infty }(\Omega )} \le 4R(1+R)$. We thus have built an admissible $(\tilde{\phi }_\rho )_\rho \in \textrm{L}^\infty (\mathbb {P}; W^{1,\infty }(\Omega ))$ from an admissible $(\phi _\rho )_\rho \in \textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega ))$ that satisfies (14), which shows that we can run the supremum $\mathrm {(D)_\mathbb {P}}'$ only over $\textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$ instead of $\textrm{L}^1(\mathbb {P}; \mathcal {C}(\Omega ))$

Existence of a maximizer There now remains to show that the supremum in $\mathrm {(D)_\mathbb {P}}'$ can be replaced by a maximum. Let $\left( (\phi _\rho ^n)_\rho \right) _{n\ge 0}$ be a maximizing sequence to $\mathrm {(D)_\mathbb {P}}'$, and assume from what precedes that this sequence belongs to $\textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$ and satisfies for all $n\ge 0$ and $\rho \in \mathcal {P}(\Omega )$, $\left\| \phi ^n_\rho \right\| _{W^{1, \infty }(\Omega )} \le 4R(1+R)$. Further assume that this sequence verifies for all $n \ge 1$,

$$\begin{aligned} \int _{\mathcal {P}(\Omega )} \left\langle {\big (\phi ^n_\rho \big )^c}|{\rho }\right\rangle \textrm{d}\mathbb {P}(\rho ) \ge \mathrm {(D)_\mathbb {P}}' - \frac{1}{n}. \end{aligned}$$

(15)

For any $n \ge 0$, the mapping $(\rho , x) \mapsto \phi ^n_\rho (x)$ is bounded in $\textrm{L}^2(\mathbb {P}\otimes \lambda )$ where $\lambda$ denotes the Lebesgue measure over $\Omega$. Therefore, by Banach–Alaoglu theorem, the sequence $\left( (\phi _\rho ^n)_\rho \right) _{n\ge 0}$ (seen as a sequence in $\textrm{L}^2(\mathbb {P}\otimes \lambda )$) admits a weakly converging subsequence in $\textrm{L}^2(\mathbb {P}\otimes \lambda )$, that we do not relabel and for which we denote $(\phi ^\infty _\rho )_\rho$ the weak limit in $\textrm{L}^2(\mathbb {P}\otimes \lambda )$. Using now Mazur’s lemma [11, Corollary 3.8], we know that there exists a sequence of integers $(N_n)_{n \ge 0}$ and coefficients $((\lambda _{n,k})_{n \le k \le N_n})_{n \ge 0} \ge 0$ satisfying for all $n \ge 0$, $\sum _{k=n}^{N_n} \lambda _{n, k} = 1$ such that the sequence $\left( (\bar{\phi }_\rho ^n)_\rho \right) _{n\ge 0}$ defined for all $n \ge 0$ and $\rho \in \mathcal {P}(\Omega )$ by $\bar{\phi }_\rho ^n:= \sum _{k=n}^{N_n} \lambda _{n,k} \phi _\rho ^k$ converges strongly to $(\phi ^\infty _\rho )_\rho$ in $\textrm{L}^2(\mathbb {P}\otimes \lambda )$. By concavity of the c-transform operation and equation (15), we then have the bound

$$\begin{aligned} \int _{\mathcal {P}(\Omega )} \left\langle {\big (\bar{\phi }^n_\rho \big )^c}|{\rho }\right\rangle \textrm{d}\mathbb {P}(\rho )&\ge \sum _{k = n}^{N_n} \lambda _{n,k} \int _{\mathcal {P}(\Omega )} \left\langle {\big (\phi ^k_\rho \big )^c}|{\rho }\right\rangle \textrm{d}\mathbb {P}(\rho ) \nonumber \\&\ge \sum _{k = n}^{N_n} \lambda _{n,k} \left( \mathrm {(D)_\mathbb {P}}' - \frac{1}{k} \right) \nonumber \\&\ge \mathrm {(D)_\mathbb {P}}' - \frac{1}{n}. \end{aligned}$$

(16)

The sequence $\left( (\bar{\phi }_\rho ^n)_\rho \right) _{n\ge 0}$ is therefore also a maximizing sequence of $\mathrm {(D)_\mathbb {P}}'$ and it also satisfies for any $n \ge 0$ and $\rho \in \mathcal {P}(\Omega )$ the bound

$$\begin{aligned} \left\| \bar{\phi }^n_\rho \right\| _{W^{1, \infty }(\Omega )} \le 4R(1+R). \end{aligned}$$

(17)

Since the sequence $\left( (\bar{\phi }_\rho ^n)_\rho \right) _{n\ge 0}$ strongly converges to $(\phi ^\infty _\rho )_\rho$ in $\textrm{L}^2(\mathbb {P}\otimes \lambda )$, one can extract a subsequence (that we do not relabel) such that for $\mathbb {P}$-almost-every $\rho \in \mathcal {P}(\Omega )$, the sequence $(\bar{\phi }^n_\rho )_{n \ge 0}$ converges to $\phi ^\infty _\rho$ in $\textrm{L}^2(\lambda )$. Using (17) and Arzelà-Ascoli theorem, we deduce that for $\mathbb {P}$-almost-every $\rho \in \mathcal {P}(\Omega )$, the sequence $(\bar{\phi }^n_\rho )_{n \ge 0}$ converges uniformly to $\phi ^\infty _\rho$ in $\mathcal {C}(\Omega )$ and that

$$\begin{aligned} \left\| \phi ^\infty _\rho \right\| _{W^{1, \infty }(\Omega )} \le 4R(1+R). \end{aligned}$$

In particular, $(\phi ^\infty _\rho )_\rho$ belongs to $\textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$ and we have the limit

$$\begin{aligned} 0 =\int _{\mathcal {P}(\Omega )} \bar{\phi }^n_\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ) \xrightarrow [n \rightarrow \infty ]{} \int _{\mathcal {P}(\Omega )} \phi ^\infty _\rho (\cdot ) \textrm{d}\mathbb {P}(\rho ), \end{aligned}$$

so that $(\phi ^\infty _\rho )_\rho$ is admissible to $\mathrm {(D)_\mathbb {P}}'$. Eventually, for $\mathbb {P}$-almost-every $\rho \in \mathcal {P}(\Omega )$, we have the limit

$$\begin{aligned} \left\langle {\big (\bar{\phi }^n_\rho \big )^c}|{\rho }\right\rangle \xrightarrow [n \rightarrow \infty ]{} \left\langle {\big (\phi ^\infty _\rho \big )^c}|{\rho }\right\rangle , \end{aligned}$$

(18)

so that by Lebesgue’s dominated convergence theorem and the bound (16),

$$\begin{aligned} \int _{\mathcal {P}(\Omega )} \left\langle {\big (\phi ^\infty _\rho \big )^c}|{\rho }\right\rangle \textrm{d}\mathbb {P}(\rho ) = \lim _{n \rightarrow +\infty } \int _{\mathcal {P}(\Omega )} \left\langle {\big (\bar{\phi }^n_\rho \big )^c}|{\rho }\right\rangle \textrm{d}\mathbb {P}(\rho ) = \mathrm {(D)_\mathbb {P}}', \end{aligned}$$

which proves that $(\phi ^\infty _\rho )_\rho \in \textrm{L}^\infty (\mathbb {P}; W^{1, \infty }(\Omega ))$ is a maximizer for $\mathrm {(D)_\mathbb {P}}'$.$\square$

Appendix B. Strong-convexity of $\mathcal {K}_\rho$ for measures with non-convex support

This section gathers occurrences of measures $\rho$ where the strong convexity estimate (4) of Assumption 1.3 is verified.

1.1 B.1 Measures with convex support

This result is mostly extracted from [19].

Proposition B.1

Let $\rho \in \mathcal {P}_{a.c.}(\Omega )$. Assume that $\textrm{spt}(\rho )$ is convex and that there exists $m_\rho , M_\rho \in (0, +\infty )$ such that $m_\rho \le \rho \le M_\rho$ on $\textrm{spt}(\rho )$. Let $\psi , \tilde{\psi } \in \mathcal {C}(\Omega )$. Then

$$\begin{aligned} \langle \psi - \tilde{\psi } | (\nabla \psi ^*)_\# \rho \rangle + C_{d, R, m_\rho , M_\rho } \mathbb {V}\textrm{ar}_{\rho }(\tilde{\psi }^* - \psi ^*) \le \mathcal {K}_\rho (\tilde{\psi }) - \mathcal {K}_\rho (\psi ), \end{aligned}$$

where $C_{d,R, m_\rho , M_\rho } = \left( e(d+1)2^{d+1} R \textrm{diam}(\textrm{spt}(\rho )) \left( \frac{M_\rho }{m_\rho } \right) ^2 \right) ^{-1}$.

Proof

We only present here a formal sketch of the proof, which heavily relies on computations done in Section 2 of [19]. Assuming that $\psi$ and $\tilde{\psi }$ are smooth enough (see Proposition 2.4 of [19]) and introducing for $t \in [0,1], \psi ^t = (1-t) \psi + t \tilde{\psi }$, Proposition 2.2 of [19] allows to differentiate $\mathcal {K}_\rho (\psi ^t)$ with respect to t and to obtain:

$$\begin{aligned} \mathcal {K}_\rho (\tilde{\psi })&- \mathcal {K}_\rho (\psi ) = \frac{\textrm{d}}{\textrm{d}t} \mathcal {K}_\rho (\psi ^t) \Big \vert _{t=0} + \int _0^1 \int _0^s \frac{\textrm{d}^2}{\textrm{d}t^2} \mathcal {K}_\rho (\psi ^t) \textrm{d}t \textrm{d}s \nonumber \\&= \langle \psi - \tilde{\psi } | (\nabla \psi ^*)_\# \rho \rangle + \int _0^1 \int _0^s \int _\Omega \langle \nabla v(\nabla (\psi ^t)^*) | \textrm{D}^2 (\psi ^t)^* \cdot \nabla v(\nabla (\psi ^t)^*)\rangle \textrm{d}\rho \textrm{d}t \textrm{d}s, \end{aligned}$$

(19)

were $v = \tilde{\psi } - \psi$. Reasoning as in the proof of Proposition 2.4 of [19], the Brascamp–Lieb concentration inequality [9] and the log-concavity of the determinant seen as an application on the set of s.d.p. matrices ensure the following bound:

$$\begin{aligned} C_{R, m_\rho , M_\rho } \min (t, 1-t)^d 2 \mathbb {V}\textrm{ar}_{\frac{1}{2}(\mu + \tilde{\mu })}(\tilde{\psi } - \psi ) \le \int _\Omega \langle \nabla v(\nabla (\psi ^t)^*) | \textrm{D}^2 (\psi ^t)^* \cdot \nabla v(\nabla (\psi ^t)^*)\rangle \textrm{d}\rho , \end{aligned}$$

where $C_{R, m_\rho , M_\rho } = \left( e R \textrm{diam}(\textrm{spt}(\rho )) \left( \frac{M_\rho }{m_\rho } \right) ^2 \right) ^{-1}$, $\mu = (\nabla \psi ^*)_\# \rho$ and $\tilde{\mu } = (\nabla \tilde{\psi })_\# \rho$. Back to (19), this leads to

$$\begin{aligned} \langle \psi - \tilde{\psi } | (\nabla \psi ^*)_\# \rho \rangle + C_{d, R, m_\rho , M_\rho } 2 \mathbb {V}\textrm{ar}_{\frac{1}{2}(\mu + \tilde{\mu })}(\tilde{\psi } - \psi ) \le \mathcal {K}_\rho (\tilde{\psi }) - \mathcal {K}_\rho (\psi ), \end{aligned}$$

where $C_{d,R, m_\rho , M_\rho } = \left( e(d+1)2^{d+1} R \textrm{diam}(\textrm{spt}(\rho )) \left( \frac{M_\rho }{m_\rho } \right) ^2 \right) ^{-1}$. We conclude using the convex analysis argument of Proposition 3.1 from [19], which directly ensures

$$\begin{aligned} \mathbb {V}\textrm{ar}_{\rho }(\tilde{\psi }^* - \psi ^*) \le 2 \mathbb {V}\textrm{ar}_{\frac{1}{2}(\mu + \tilde{\mu })}(\tilde{\psi } - \psi ). \end{aligned}$$

We get the general case (without the smoothness assumptions on $\psi$ and $\tilde{\psi }$) using approximation arguments presented in Proposition 2.5 and 2.7 of [19].

1.2 B.2 Measures with connected union of convex sets as support

We extend Proposition B.1 to the case of a source measure $\rho$ with a possibly non-convex support. We will assume that $\textrm{spt}(\rho )$ can be written as a connected finite union of convex sets.

Proposition B.2

Let $\rho \in \mathcal {P}_{a.c.}(\Omega )$ such that there exists $m_\rho , M_\rho \in (0, +\infty )$ verifying $m_\rho \le \rho \le M_\rho$ on $\textrm{spt}(\rho )$. Assume that $\textrm{spt}(\rho )$ is connected and that there exists $N\ge 1$ convex sets $(C_i)_{1 \le i \le N}$ in $\Omega$ such that $\textrm{spt}(\rho ) = \bigcup _{i=1}^N C_i$. Also assume that for any $i \ne j$ such that $C_i \cap C_j \ne \emptyset$, one has $\rho (C_i \cap C_j) > 0$. Then there exists a constant $c_\rho$ depending on $\rho$ such that for any $\psi , \tilde{\psi } \in \mathcal {C}(\Omega )$,

$$\begin{aligned} \langle \psi - \tilde{\psi } | (\nabla \psi ^*)_\# \rho \rangle + c_\rho \mathbb {V}\textrm{ar}_{\rho }(\tilde{\psi }^* - \psi ^*) \le \mathcal {K}_\rho (\tilde{\psi }) - \mathcal {K}_\rho (\psi ). \end{aligned}$$

Remark B.1

(Constant $c_\rho$ and Poincaré–Wirtinger constant of $\rho$) The constant $c_\rho$ of Proposition B.2 is not made precise in the statement. A look at the proof of this proposition only allows to bound $c_\rho$ in terms of the second smallest eigenvalue $\lambda _2(L)$ of a weighted graph Laplacian L, that is built from the graph whose vertices are the convex sets $C_i$ and whose edge weights are the masses $\rho (C_i \cap C_j)$ that $\rho$ grants to the intersection of the convex sets $C_i$ and $C_j$. The constant $c_\rho$ then reads:

$$\begin{aligned} c_\rho = \left( e(d+1)2^{d+1} R^2 \left( \frac{M_\rho }{m_\rho } \right) ^2 \left( N^2 + \frac{ 2 N^3}{\lambda _2(L)} \right) \right) ^{-1}. \end{aligned}$$

The quantity $\lambda _2(L)$ is not explicit, but it can be linked to the weighted Cheeger constant of $\rho$, defined by

$$\begin{aligned} h(\rho ) = \inf _{A \subset \textrm{spt}(\rho )} \frac{ \left| \partial A\right| _\rho }{ \min (\rho (A), \rho (\textrm{spt}(\rho ) \setminus A)) }, \end{aligned}$$

where $\left| \partial A\right| _\rho = \int _{\partial A \cap \textrm{int}(\textrm{spt}(\rho ))} \rho (x) \textrm{d}\mathcal {H}^{d-1}(x)$ and where the infimum is taken over Lipschitz domains $A \subset \textrm{int}(\textrm{spt}(\rho ))$ with boundary of finite $\mathcal {H}^{d-1}$-measure. Quoting [25] (Lemma 5.3), this constant can in turn be linked to the $\textrm{L}^1$ Poincaré–Wirtinger constant $C_{PW}(\rho )$ of $\rho$. Indeed, $h(\rho )$ is positive whenever $\rho$ satisfies an $\textrm{L}^1$ Poincaré–Wirtinger inequality, i.e. whenever there exists a finite $C_{PW}(\rho ) > 0$ such that for all smooth function f on $\Omega$,

$$\begin{aligned} \left\| f - \mathbb {E}_\rho f\right\| _{\textrm{L}^1(\rho )} \le C_{PW}(\rho ) \left\| \nabla f\right\| _{\textrm{L}^1(\rho ; \mathbb {R}^d)}. \end{aligned}$$

The Poincaré–Wirtinger constant $C_{PW}(\rho )$ and the Cheeger constant $h(\rho )$ are then related by the inequality

$$\begin{aligned} h(\rho ) \ge \frac{2}{C_{PW}(\rho )}. \end{aligned}$$

Using ideas similar to the ones found in Section 5.2 of [25], the eigenvalue $\lambda _2(L)$ can be bounded in terms of the Cheeger constant of $\rho$, and thus in terms of $C_{PW}(\rho )$. We do not detail this comparison here but only report that $c_\rho$ may be written

$$\begin{aligned} c_\rho = \left( e(d+1)2^{d+1} R^2 \left( \frac{M_\rho }{m_\rho } \right) ^2 N \left( N + \frac{1}{2}\left( \frac{ M_\rho s_{d-1} R^{d-1} N^2 C_{PW}(\rho )}{ \varepsilon ^2 }\right) ^3 \right) \right) ^{-1}, \end{aligned}$$

where $s_{d-1}$ denotes the surface area of the unit sphere in $\mathbb {R}^d$ and

$$\begin{aligned} \varepsilon = \min \left( \min _{i, j \vert C_i \cap C_j \ne \emptyset } \rho (C_i \cap C_j), \min _i \rho \left( C_i \setminus \cup _{j \ne i} C_j \right) \right) > 0. \end{aligned}$$

Proof of Proposition B.2

Let’s denote for now $f = \tilde{\psi }^* - \psi ^*$. We will first exploit a discrete Laplacian over $\mathcal {X}= \textrm{spt}(\rho )$ in order to upper bound $\mathbb {V}\textrm{ar}_\rho (f)$ by a sum of variances of f w.r.t. probability measures supported over the convex sets $(C_i)_i$. We will then use Proposition B.1 to conclude.

For any $i \in \{1,\dots ,N\}$, we denote $\rho _i = \frac{1}{\rho (C_i)} \rho _{\vert C_i}$ and $m_i = \int _{C_i} f \textrm{d}\rho _i$. Then one has the following bound:

$$\begin{aligned} \mathbb {V}\textrm{ar}_\rho (f)&= \frac{1}{2} \int _{\mathcal {X}\times \mathcal {X}} (f(x) - f(y))^2 \textrm{d}\rho (x) \textrm{d}\rho (y) \nonumber \\&\le \frac{1}{2} \sum _{i,j} \int _{C_i \times C_j} (f(x) - f(y))^2 \textrm{d}\rho (x) \textrm{d}\rho (y) \nonumber \\&= \frac{1}{2} \sum _{i,j} \int _{C_i \times C_j} (f(x) - m_i + m_i - m_j + m_j - f(y))^2 \textrm{d}\rho (x) \textrm{d}\rho (y) \nonumber \\&= \left( \sum _i \rho (C_i) \right) \sum _i \int _{C_i} (f(x) - m_i)^2 \textrm{d}\rho (x) + \frac{1}{2} \sum _{i,j} (m_i - m_j)^2 \rho (C_i) \rho (C_j) \nonumber \\&= \left( \sum _i \rho (C_i) \right) \sum _i \rho (C_i) \mathbb {V}\textrm{ar}_{\rho _i}(f) + \frac{1}{2} \sum _{i,j} (m_i - m_j)^2 \rho (C_i) \rho (C_j). \end{aligned}$$

(20)

We now consider the graph $G = (\{C_i\}_{1 \le i \le N}, \{w_{ij}\}_{1 \le i,j \le N})$ with vertices $\{C_i\}_{1 \le i \le N}$ and weighted edges $\{w_{ij}\}_{1 \le i,j \le N}$ defined by

$$\begin{aligned} \forall i,j \in \{1, \dots , N\}, \quad w_{ij} = \rho (C_i \cap C_j). \end{aligned}$$

By construction, this graph has a single connected component. We introduce the weighted Laplacian matrix $L \in \mathbb {R}^{N\times N}$ of G as follows:

$$\begin{aligned} \forall i,j \in \{1, \dots , N\}, \quad L_{ij} = \left\{ \begin{array}{ll} \sum _{k} w_{ik} &{} \text{ if } i = j, \\ -w_{ij} &{} \text{ else. } \end{array} \right. \end{aligned}$$

Then L is a symmetric and positive semi-definite matrix. Its null space is made of constant vectors and we denote $\lambda _2(L)$ its second smallest eigenvalue, which is non-zero. Denoting $m = (m_i)_{1\le i \le N} \in \mathbb {R}^N$, we introduce $\bar{m} = \left( \frac{1^{}}{N} \sum _i m_i\right) \mathbbm {1}_N \in \mathbb {R}^N$ the constant vector whose coordinates equal the mean of m (we use $\mathbbm {1}_N = (1)_{1\le i \le N} \in \mathbb {R}^N$). Notice that $m - \bar{m}$ is in the orthogonal to the null space of L, ensuring the following bound:

$$\begin{aligned} \frac{1}{2} \sum _{i,j} (m_i - m_j)^2 \rho (C_i) \rho (C_j)&\le N^2 \frac{1}{2} \sum _{i,j} (m_i - m_j)^2 \frac{1}{N^2} \nonumber \\&= N \left\| m - \bar{m} \right\| ^2 \nonumber \\&\le \frac{ N }{\lambda _2(L)} \langle m - \bar{m} | L \left( m - \bar{m}\right) \rangle \nonumber \\&= \frac{ N }{\lambda _2(L)} \sum _{i, j} w_{ij} (m_i^2 - m_i m_j) \nonumber \\&= \frac{ N }{\lambda _2(L)} \sum _{i, j} \frac{w_{ij}}{2} (m_i - m_j)^2. \end{aligned}$$

(21)

But for any i, j such that $w_{ij}>0$, denoting $m_{i \cap j} = \frac{1}{\rho (C_i \cap C_j)} \int _{C_i \cap C_j} f \textrm{d}\rho$, one has

$$\begin{aligned} \frac{1}{2} (m_i - m_j)^2 \le (m_{i \cap j} - m_i)^2 + (m_{i \cap j} - m_j)^2. \end{aligned}$$

And for such i, j,

$$\begin{aligned} (m_{i \cap j} - m_i)^2&= \left( \frac{1}{\rho (C_i \cap C_j)} \int _{C_i \cap C_j} (f - m_i) \textrm{d}\rho \right) ^2 \\&\le \frac{1}{\rho (C_i \cap C_j)} \int _{C_i} (f - m_i)^2 \textrm{d}\rho \\&= \frac{\rho (C_i)}{w_{ij}} \mathbb {V}\textrm{ar}_{\rho _i}(f), \end{aligned}$$

where we used Jensen’s inequality and the fact that $C_i \cap C_j \subset C_i$. A similar bound can be shown for $(m_{i \cap j} - m_j)^2$, and plugging these into (21) yields

$$\begin{aligned} \frac{1}{2} \sum _{i,j} (m_i - m_j)^2 \rho (C_i) \rho (C_j)&\le \frac{ N }{\lambda _2(L)} \sum _{i} \sum _{j \vert C_i \cap C_j \ne \emptyset } \left( \rho (C_i) \mathbb {V}\textrm{ar}_{\rho _i}(f) + \rho (C_j) \mathbb {V}\textrm{ar}_{\rho _j}(f)\right) \\&\le \frac{ 2 N^2 }{\lambda _2(L)} \sum _{i} \rho (C_i) \mathbb {V}\textrm{ar}_{\rho _i}(f). \end{aligned}$$

Injecting this into (20) yields

$$\begin{aligned} \mathbb {V}\textrm{ar}_\rho (f) \le \left( N + \frac{ 2 N^2}{\lambda _2(L)} \right) \sum _i \rho (C_i) \mathbb {V}\textrm{ar}_{\rho _i}(f). \end{aligned}$$

(22)

Now recalling that $f = \psi - \tilde{\psi }$, we have by Proposition B.1 for any $i\in \{1, \dots , N\}$ that

$$\begin{aligned} \langle \psi - \tilde{\psi } | (\nabla \psi ^*)_\# \rho _i\rangle + C_{d,R, m_\rho , M_\rho } \mathbb {V}\textrm{ar}_{\rho _i}(\tilde{\psi }^* - \psi ^*) \le \mathcal {K}_{\rho _i}(\tilde{\psi }) - \mathcal {K}_{\rho _i}(\psi ), \end{aligned}$$

where $C_{d,R, m_\rho , M_\rho }= \left( e(d+1)2^{d+1} R^2 \left( \frac{M_\rho }{m_\rho } \right) ^2 \right) ^{-1}$. Weighting this last inequality with $\rho (C_i)$ and summing over $i \in \{1, \dots , N\}$, this raises

$$\begin{aligned} \langle \psi - \tilde{\psi } | (\nabla \psi ^*)_\# \rho \rangle + \frac{C_{d,R, m_\rho , M_\rho }}{N} \sum _{i=1}^N \rho (C_i) \mathbb {V}\textrm{ar}_{\rho _i}(\tilde{\psi }^* - \psi ^*) \le \mathcal {K}_{\rho }(\tilde{\psi }) - \mathcal {K}_{\rho }(\psi ). \end{aligned}$$

Using (22) eventually gives

$$\begin{aligned} \langle \psi - \tilde{\psi } | (\nabla \psi ^*)_\# \rho \rangle + c_{\rho } \mathbb {V}\textrm{ar}_{\rho }(\tilde{\psi }^* - \psi ^*) \le \mathcal {K}_{\rho }(\tilde{\psi }) - \mathcal {K}_{\rho }(\psi ), \end{aligned}$$

where $c_\rho = \left( e(d+1)2^{d+1} R^2 \left( \frac{M_\rho }{m_\rho } \right) ^2 \left( N^2 + \frac{ 2 N^3}{\lambda _2(L)} \right) \right) ^{-1}$.