Optimal transport metrics

Wasserstein distances, Monge-Kantorovich metrics, Earthmover distances


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.

I am about to do a reading group based on Peyré’s course, so will be harmonising the notation with that soon so I can use this notebook to assist.

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.

Practically, one usually sees \(p\in\{1,2\}\). For \(p=1\) then

\[W_1(P,Q)=\inf_{\gamma \in \Pi( P , Q )}\mathbb{E}_{(x,y)\sim \gamma}\left[d(x,y)\right]\]

This 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 you are about such an intractable distance? Because gives good error bounds. We know that if \(W_p(\nu\hat{nu}) \leq \epsilon\), then for any L-Lipschitz function \(\phi\), \(|\nu(\phi) - \hat{\nu}(\phi)| \leq L\epsilon.\) See (J. H. Huggins et al. 2018b, 2018a) for some specifics.

Analytic expressions

Gaussian

Useful: Two Gaussians may be related thusly (Givens and Shortt 1984) for a Wasserstein-2 \(W_2(\mu;\nu):=\inf\mathbb{E}(\Vert X-Y\Vert_2^2)^{1/2}\) for \(X\sim\nu\), \(Y\sim\mu\).

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

Kontorovich-Rubinstein duality

Vincent Hermann gives an excellent practical introduction.

“Neural Net distance”

Wasserstein distance with a baked in notion of the capacity of the function class which approximate the true Wasserstein. (Arora et al. 2017) Is this actually used?

Fisher distance

Specifically \((p,\nu)\)-Fisher distances, in the terminology of (J. H. Huggins et al. 2018b). They use these distances as a computationally tractable proxy (in fact, bound) for Wasserstein distances during inference. See Fisher distances.

Sinkhorn divergence

A regularised version of a Wasserstein metric. (Cuturi 2013)

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

Here \(H\) is the entropy.

In practice this seems 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. (Altschuler et al. 2019; Blanchet et al. 2018)

TBD.

Awaiting filing

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