Optimal transport distances

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.

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 \(p=1\) one uses \[W_1(P,Q)=\inf_{\gamma \in \Pi( P , Q )}\mathbb{E}_{(x,y)\sim \gamma}\left[d(x,y)\right]\]

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 it bounds the errors from approximate distributions.

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 (Huggins et al. 2018) for some specifics.

Analytic expressions

Gaussian

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

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

((Givens and Shortt 1984))

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 (Huggins et al. 2018). They use these distances as a computationally tractable proxy for Wasserstein distances. See Fisher distances.

Awaiting filing

Fundamental theorem of optimal transport.

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