Asymptotic equivalence of W2 and H^-1
Motivation
The quadratic Wasserstein distance and \(\dot{H}^{-1}\) distance become asymptotically equivalent when the when the measures are absolutely continuous with respect to Lebesgue measure with density close to the value \(\varrho = 1\). This is particularly of interest since the space \(H^{-1}\) is a Hilbert space as opposed to \(W_2\) being only a metric space. This allows one to extend several well-known results about continuity of various operators in \(H^{-1}\) to \(W_2\) by asymptotic equivalence. This equivalence is also important numerically, where computing \(H^{-1}\) is much easier than computing \(W_2\).
Furthermore, this asymptotic equivalence is relevant for evolution problems with the constraint \(\varrho \leq 1\), such as crowd motion. 1
Formalization
Definition of \(\dot{H}^{-1}\)
The negative Sobolev norm \(\| \cdot \|_\dot{H}^{-1}\) is defined 2 3 to be
\(\| \mu - \nu \|_{\dot{H}^{-1} (\Omega)} := \sup \left \{ \int_{\Omega} \phi \, \mathrm{d}( \mu - \nu ) : \phi \in C^{\infty}_c(\Omega), \, \| \nabla \phi \|_{L^2(\Omega)} \leq 1 \right \} .\)
Lemma
Let \(\mu, \nu\) be measures that are absolutely continuous with respect to Lebesgue measure on a convex domain \(\Omega\), with densities bounded above by the same constant \(C > 0\). Then, for all functions \(\phi \in H^1(\Omega)\):
\(\int_\Omega \phi \, \mathrm{d}( \mu - \nu ) \leq \sqrt{C} \| \nabla \phi \|_{L^2(\Omega)} W_2(\mu, \nu)\)
Proof of the lemma can be found Chapter 5, page 210 of 4.
\(\dot{H}^{-1}\) as a Dual
This material is adapted from 5.
An important property of \(\dot{H}^{-1}\) is its characterization as a dual, which justifies the notation. Let \(\Omega \subseteq \mathbb{R}^d\) be an open and connected subset. For \(\phi \in C^1(\Omega)\),
\(\| \phi \|_{\dot{H}^1} := \| \nabla \phi \|_{L^2(\Omega)} := \left[ \int_{\Omega} | \nabla \phi(x) |^2 \, \mathrm{d}x \right]^{\frac{1}{2}}\)
defines a semi-norm. Then for an absolutely continuous signed measure on \(\Omega\) with zero total mass,
\(\| \nu \|_{ \dot{H}^{-1} } := \sup \left \{ | \langle \phi , \nu \rangle | : \phi \in C^1(\Omega) , \, \| \phi \|_{\dot{H}^1} \leq 1 \right \} = \sup \left \{ \left| \int_{\Omega} \phi(x) \, \mathrm{d}\nu(x) \right| : \phi \in C^1(\Omega) , \, \| \phi \|_{\dot{H}^1} \leq 1 \right \} .\)
The space \(\dot{H}^{-1}\) is the dual space of zero-mean \(H^1(\Omega)\) functions endowed with the norm \(L^2\) norm on the gradient.
Theorem
Let \(\mu, \nu\) be absolutely continuous measures on a convex domain \(\Omega\), with densities bounded from below and from above by the same constants \(a, b\) with \(0 < a < b < +\infty\). Then
\(b^{-\frac{1}{2}} || \mu - \nu ||_{\dot{H}^{-1} (\Omega)} \leq W_2( \mu, \nu) \leq a^{-\frac{1}{2}}|| \mu - \nu ||_{\dot{H}^{-1}(\Omega)}\)
The proof of the theorem uses the above lemma and can be found Chapter 5, page 211 of 6.
Localization
The following material is adapted from 7.
This section deals with the problem of localization of the quadratic Wasserstein distance: if \(\mu , \nu\) are (signed) measures on \(\mathbb{R}^d\) that are close in the sense of \(W_2\), do they remain close to each other when restricted to subsets of \(\mathbb{R}^d\)?
Notation
Here we are working in Euclidean space \(\mathbb{R}^d\) with the Lebesgue measure \(\lambda\).
- Recall that for a subset \(A \subseteq\mathbb{R}^d\),
\(\mathrm{dist}(x,A) := \inf \{ |x - y| : y \in A \}\)
denotes the distance between a point \(x\) and the subset \(A\).
For a (signed) measure \(\mu\) on \(\mathbb{R}^d\) and \(\varphi : \mathbb{R}^d \to \mathbb{R}\) a nonnegative and measurable function, \(\varphi \cdot \mu\) denotes the measure such that \(\mathrm{d}(\varphi \cdot \mu) = \varphi(x) \, \mathrm{d}\mu(x)\).
The norm
\(\| \mu \|_1 := \int_{\mathbb{R}^d} \, |\mathrm{d}\mu(x)|\)
denotes the total variation norm of the signed measure \(\mu\). If \(\mu\) is in fact a measure, then \(\| \mu \|_1 = \mu ( \mathbb{R}^d )\).
Now we can ask the original question more precisely. If \(\varphi : \mathbb{R}^d \to \mathbb{R}\) is non-negative and compactly supported satisfying further technical assumptions to be specified later, we wish to bound \(W_2 ( a \varphi \cdot \mu , \varphi \cdot \nu)\) by \(W_2(\mu,\nu)\), where \(a\) is a constant factor ensuring that \(a\varphi \cdot \mu\) and \(\varphi \cdot \nu\) have the same mass. The factor of \(a\) is necessary, otherwise the \(W_2\) distance between \(\varphi \cdot \mu\) and \(\varphi \cdot \nu\) is in general not well-defined.
Theorem
Let \(\mu , \nu\) be measures on \(\mathbb{R}^d\) having the same total mass, and let \(B\) be a ball in \(\mathbb{R}^d\). Assume that on \(B\), the density of \(\mu\) with respect to the Lebesgue measure is bounded above and below, that is
\(\exists 0 < m_1 \leq m_2 < \infty \quad \forall x \in B \quad m_1 \mathrm{d}\lambda(x) \leq \mathrm{d}\mu(x) \leq m_2 \mathrm{d}\lambda(x).\)
Let \(\varphi : \mathbb{R}^d \to (0,+\infty)\) be a \(k\)-Lipschitz function for some \(0 \leq k < \infty\) supported in \(B\), and suppose that \(\varphi\) is bounded above and below by the map
\(x \mapsto \mathrm{dist}(x,B^c)^2\)
on \(B\), that is, there exists constants \(0 < c_1 \leq c_2 < \infty\) such that for all \(x \in B\),
\(c_1 \mathrm{dist}(x,B^c)^2 \leq \varphi(x) \leq c_2 \mathrm{dist}(x,B^c)^2 .\)
Then, denoting
\(a := \| \varphi \cdot \nu \|_1 / \| \varphi \cdot \mu \|_1 = \frac{ \int_{\mathbb{R}^d} |\varphi(x) \, \mathrm{d}\mu(x)| }{ \int_{\mathbb{R}^d} |\varphi(x) \, \mathrm{d}\nu(x)| } ,\)
we have
\(W_2 (a\varphi \cdot \mu , \varphi \cdot \nu) \leq C(n)^{\frac{1}{2}} \left( \frac{ c_2 m_2 }{ c_1 m_1 } \right)^{\frac{3}{2}} k c_1^{-\frac{1}{2}} W_2(\mu,\nu) ,\)
for \(C(n) < \infty\) some absolute constant depending only on \(n\). Moreover, taking \(C(n) := 2^{11} n\) fits. Furthermore, that \(\varphi\) is supported in a ball is not necessary, as it can be supported in a cube or a simplex.
The proof can be found in 8.
Connection with the Vlasov-Poisson Equation
Loeper 9 contributed an earlier result on a bound between \(W_2\) and \(\dot{H}^{-1}\) for bounded densities in studying the existence of solutions to the Vlasov-Poisson equation. Namely, Loeper proved that that if \(\rho_1 , \rho_2\) be probability measures on \(\mathbb{R}^d\) with \(L^{\infty}\) densities with respect to the Lebesgue measure. Let \(\Psi_i\), \(i = 1, 2\) solve
\(-\Delta \Psi_i = \rho_i \qquad \text{in } \mathbb{R}^d ,\)
\(\Psi_i(x) \to 0 \qquad \text{as } |x| \to \infty ,\)
in the integral sense, that is,
\(\Psi_i(x) = \frac{1}{4\pi} \int_{\mathbb{R}^d} \frac{\rho_i(y)}{|x - y|} \, \mathrm{d}y .\)
Then
\(\| \nabla \Psi_1 - \nabla \Psi_2 \|_{L^2(\mathbb{R}^d)} \leq \left[ \max \left \{ \| \rho_1 \|_{L^{\infty}} , \| \rho_2 \|_{L^{\infty}} \right \} \right]^{\frac{1}{2}} W_2(\rho_1,\rho_2) .\)
Loeper also extended the result to finite measures with the same total mass.
References
Footnotes
F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, pages 209-211↩︎
F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, pages 209-211↩︎
https://www.sciencedirect.com/science/article/pii/S0021782406000067 Loeper, Grégoire. Uniqueness of the solution to the Vlasov–Poisson system with bounded density. Journal de Mathématiques Pures et Appliquées, Volume 86, Issue 1, 2006, Pages 68-79, ISSN 0021-7824.↩︎
F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, pages 209-211↩︎
https://arxiv.org/abs/1104.4631v2 Peyre, Rémi. Comparison between \(W_2\) distance and \(\dot{H}^{-1}\) norm, and localisation of Wasserstein distance.↩︎
F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, pages 209-211↩︎
https://arxiv.org/abs/1104.4631v2 Peyre, Rémi. Comparison between \(W_2\) distance and \(\dot{H}^{-1}\) norm, and localisation of Wasserstein distance.↩︎
https://arxiv.org/abs/1104.4631v2 Peyre, Rémi. Comparison between \(W_2\) distance and \(\dot{H}^{-1}\) norm, and localisation of Wasserstein distance.↩︎
https://www.sciencedirect.com/science/article/pii/S0021782406000067 Loeper, Grégoire. Uniqueness of the solution to the Vlasov–Poisson system with bounded density. Journal de Mathématiques Pures et Appliquées, Volume 86, Issue 1, 2006, Pages 68-79, ISSN 0021-7824.↩︎