# Optimal transport metrics

Wasserstein distances, Monge-Kantorovich metrics, Earthmover distances

May 30, 2019 — June 8, 2021

approximation
functional analysis
measure
metrics
optimization
probability
statistics

danger: I am half way through editing this! notation inconsistencies and indeed plain falsehoods abound.

I presume there are other uses for optimal transport distances apart from as probability metrics, but so far I only care about them in that context, so this will be skewed that way.

Let $$(M,d)$$ be a metric space for which every probability measure on $$M$$ is a Radon measure. For $$p\ge 1$$, let $$\mathcal{P}_p(M)$$ denote the collection of all probability measures $P$ on $$M$$ with finite $$p^{\text{th}}$$ moment for some $$x_0$$ in $$M$$,

$\int_{M} d(x, x_{0})^{p} \, \mathrm{d} P (x) < +\infty.$

Then the $$p^{\text{th}}$$ Wasserstein distance between two probability measures $P$ and $Q$ in $$\mathcal{P}_p(M)$$ is defined as

$W_{p} ( P, Q ):= \left( \inf_{\gamma \in \Pi ( P, Q )} \int_{M \times M} d(x, y)^{p} \, \mathrm{d} \gamma (x, y) \right)^{1/p},$ where $$\Pi( P, Q )$$ denotes the collection of all measures on $$M \times M$$ with marginal distributions $P$ and $Q$ respectively.

We can work equivalently with RVs distributed according to these measures, respectively $$X\sim P,Y\sim Q,$$ and then we consider $$\gamma$$ to range over joint distributions for these RVs, so that $W_p(X,Y):=W_p(P;Q):=\inf_{\gamma}\mathbb{E}[d(X,Y)^p]^{1/p}$

Also, the cost is usually just the Euclidean/$$L_2$$ distance, so that \begin{aligned} W_p(X,Y):=W_p(P;Q) &:=\inf_{\gamma}\mathbb{E}[\Vert X-Y\Vert_2^p]^{1/p}. \end{aligned}

Practically, one usually sees $$p\in\{1,2\}$$. Maybe $$p=\infty$$ is interesting also . For $$p=1$$ we have an extremely useful representation in terms of optimisation over functions, the Kontorovich-Rubinstein duality. $W_1(P, Q)=\sup _{f,g:f(x)+g(y)\leq d(x,y)} \mathbb{E}_{x \sim P}[f(x)]-\mathbb{E}_{x \sim Q}[g(x)].$ For Euclidean distance this optimisation ranges over 1-Lipschitz functions, $W_1(P, Q)=\sup _{\|f\|_{L} \leq 1} \mathbb{E}_{x \sim P}[f(x)]-\mathbb{E}_{x \sim Q}[f(x)].$ TODO: check that.

The Wasserstein distance between two objects is frequently intractable, or at least has no closed form, but you can find it for some useful special cases, or bound/approximate it in others.

🏗 discuss favourable properties of this metric (triangle inequality, bounds on moments etc).

But why do we are about such an intractable distance? Because it gives good error bounds for sufficiently nice expectations. We know that if $$W_p(X,Y) \leq \epsilon$$, then for any L-Lipschitz function $$f$$, $$|f(X) - f(Y)| \leq L\epsilon.$$

## 1 Analytic expressions

### 1.1 Gaussian

Useful: Two Gaussians may be related thusly for a Wasserstein-2 for $$X\sim\nu$$, $$Y\sim\mu$$.

\begin{aligned} d&:= W_2(\mathcal{N}(m_1,\Sigma_1);\mathcal{N}(m_2,\Sigma_2))\\ \Rightarrow d^2&= \Vert m_1-m_2\Vert_2^2 + \mathrm{Tr}(\Sigma_1+\Sigma_2-2(\Sigma_1^{1/2}\Sigma_2\Sigma_1^{1/2})^{1/2}). \end{aligned}

### 1.2 Any others?

None that I know.

## 2 Kontorovich-Rubinstein duality

Vincent Hermann gives an excellent practical introduction.

## 3 “Neural Net distance”

Wasserstein distance with a baked in notion of the capacity of the function class which approximate the true Wasserstein . Is this actually used?

## 4 Fisher distance

Specifically $$(p,\nu)$$-Fisher distances, in the terminology of . They use these distances as a computationally tractable proxy (in fact, bound) for Wasserstein distances during inference. See Fisher distances.

## 5 Sinkhorn divergence

A regularised version of a Wasserstein metric .

$W_{p,\eta} ( P, Q )^p:= \inf_{\gamma \in \Pi ( P, Q )} \int_{M \times M} d(x, y)^{p} \, \mathrm{d} \gamma (x, y) -H(\gamma).$

Here $$H$$ is the entropy.

The examples I saw seem to be only applied to measures over finite sets (i.e. histograms, weighted point sets), where there are many neat tricks to make calculations tractable. Can it be more general?

TBD.

## 6 Introductions

have been recommended to me as compact modern introductions.

Peyré’s course to accompany Peyré and Cuturi (2019) has been recommended to me and comes with course notes.

Or the bibliography in POT: Python Optimal Transport.

## 8 Connection to Maximum mean discrepancy

Feydy et al. (2019) connects OT to maximum mean discrepance.

## 9 Incoming

Michele Coscia’s paper using a graph Laplacian to calculate an approximate Earth mover distance over a graph topology. Buzzword use case: interpretably inferring graph transmission rate of a disease. This looks simple; surely it must be a known result in optimal transport metric studies?

## 10 References

Altschuler, Bach, Rudi, et al. 2019. In Advances in Neural Information Processing Systems 32.
Ambrosio, and Gigli. 2013. In Modelling and Optimisation of Flows on Networks: Cetraro, Italy 2009, Editors: Benedetto Piccoli, Michel Rascle. Lecture Notes in Mathematics.
Ambrosio, Gigli, and Savare. 2008. Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics. ETH Zürich.
Angenent, Haker, and Tannenbaum. 2003. SIAM Journal on Mathematical Analysis.
Arjovsky, Chintala, and Bottou. 2017. In International Conference on Machine Learning.
Arora, Ge, Liang, et al. 2017. arXiv:1703.00573 [Cs].
Arras, Azmoodeh, Poly, et al. 2017. arXiv:1704.01376 [Math].
Bachem, Lucic, and Krause. 2017. arXiv Preprint arXiv:1703.06476.
Bion-Nadal, and Talay. 2019. The Annals of Applied Probability.
Blanchet, Chen, and Zhou. 2018. arXiv:1802.04885 [Stat].
Blanchet, Jambulapati, Kent, et al. 2018. arXiv:1810.07717 [Cs].
Blanchet, Kang, and Murthy. 2016. arXiv:1610.05627 [Math, Stat].
Blanchet, Murthy, and Si. 2019. arXiv:1906.01614 [Math, Stat].
Bolley, Gentil, and Guillin. 2012. Journal of Functional Analysis.
Bolley, and Villani. 2005. Annales de La Faculté Des Sciences de Toulouse Mathématiques.
Canas, and Rosasco. 2012. arXiv:1209.1077 [Cs, Stat].
Carlier, Cuturi, Pass, et al. 2017. “Optimal Transport Meets Probability, Statistics and Machine Learning.”
Carlier, Galichon, and Santambrogio. 2008.
Champion, De Pascale, and Juutinen. 2008. SIAM Journal on Mathematical Analysis.
Chizat, and Bach. 2018. In Proceedings of the 32nd International Conference on Neural Information Processing Systems. NIPS’18.
Chizat, Oyallon, and Bach. 2018. arXiv:1812.07956 [Cs, Math].
Chu, Blanchet, and Glynn. 2019. In ICML.
Coscia. 2020. “Generalized Euclidean Measure to Estimate Network Distances.”
Cuturi. 2013. In Advances in Neural Information Processing Systems 26.
Dowson, and Landau. 1982. Journal of Multivariate Analysis.
Dudley. 2002. Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74.
Fatras, Zine, Flamary, et al. 2020. In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics.
Fernholz. 1983. von Mises calculus for statistical functionals. Lecture Notes in Statistics 19.
Feydy, Séjourné, Vialard, et al. 2019. In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics.
Frogner, Zhang, Mobahi, et al. 2015. In Advances in Neural Information Processing Systems 28.
Gao, and Kleywegt. 2022.
Givens, and Shortt. 1984. The Michigan Mathematical Journal.
Gozlan, and Léonard. 2010. arXiv:1003.3852 [Math].
Guo, Hong, Lin, et al. 2017. arXiv:1705.07164 [Cs, Stat].
Henry-Labordère. 2017. Model-Free Hedging: A Martingale Optimal Transport Viewpoint. Chapman and Hall CRC Financial Mathematics 1.0.
Huggins, Jonathan, Adams, and Broderick. 2017. In Advances in Neural Information Processing Systems 30.
Huggins, Jonathan H., Campbell, Kasprzak, et al. 2018a. arXiv:1806.10234 [Cs, Stat].
———, et al. 2018b. arXiv:1809.09505 [Cs, Math, Stat].
Ji, and Liang. 2018.
Khan, and Zhang. 2022. Information Geometry.
Kloeckner. 2021. Mathematical Proceedings of the Cambridge Philosophical Society.
Knott, and Smith. 1984. Journal of Optimization Theory and Applications.
Kontorovich, and Raginsky. 2016. arXiv:1602.00721 [Cs, Math].
Léonard. 2014. Discrete & Continuous Dynamical Systems - A.
Liu, Huidong, Gu, and Samaras. 2018. In International Conference on Machine Learning.
Liu, Qiang, and Wang. 2019. In Advances In Neural Information Processing Systems.
Mahdian, Blanchet, and Glynn. 2019. arXiv:1906.03317 [Cs, Math, Stat].
Mallasto, Gerolin, and Minh. 2021. Information Geometry.
Marzouk, Moselhy, Parno, et al. 2016. In Handbook of Uncertainty Quantification.
Maurya. 2018. “Optimal Transport in Statistical Machine Learning : Selected Review and Some Open Questions.” In.
Mohajerin Esfahani, and Kuhn. 2018. Mathematical Programming.
Montavon, Müller, and Cuturi. 2016. In Advances in Neural Information Processing Systems 29.
Olkin, and Pukelsheim. 1982. Linear Algebra and Its Applications.
Panaretos, and Zemel. 2019. Annual Review of Statistics and Its Application.
———. 2020. An Invitation to Statistics in Wasserstein Space. SpringerBriefs in Probability and Mathematical Statistics.
Peyre. 2019. “Course Notes on Computational Optimal Transport.”
Peyré, and Cuturi. 2019. Computational Optimal Transport.
Piccoli, and Rossi. 2016. Archive for Rational Mechanics and Analysis.
Santambrogio. 2015. Optimal Transport for Applied Mathematicians. Edited by Filippo Santambrogio. Progress in Nonlinear Differential Equations and Their Applications.
Singh, and Póczos. 2018. arXiv:1802.08855 [Cs, Math, Stat].
Solomon, de Goes, Peyré, et al. 2015. ACM Transactions on Graphics.
Spantini, Bigoni, and Marzouk. 2017. Journal of Machine Learning Research.
Takatsu. 2008.
Thorpe. 2018. “Introduction to Optimal Transport.”
Verdinelli, and Wasserman. 2019. Electronic Journal of Statistics.
Villani. 2009. Optimal Transport: Old and New. Grundlehren Der Mathematischen Wissenschaften.
Weed, and Bach. 2017. arXiv:1707.00087 [Math, Stat].
Zemel. 2012. “Optimal Transportation: Continuous and Discrete.”
Zhu, Jiao, and Tse. 2020. IEEE Transactions on Information Theory.