Wasserstein Distance
Definition
Consider, general functions \(f\) and \(g\), the Wasserstein distance is \[\min_{\text{all map }T}\{\sum_{\text{all movements of }T}\text{distance moved}\times\text{amount moved}\}\] For \(f:X\rightarrow{}R^+,g:Y\rightarrow{}R^+\), the distance can be formulated as \[W_p(f,g)=\left(\inf_{T\in\mathcal{M}}\int{}|x-T(x)|^pf(x)dx\right)^{1/p}\] where \(\mathcal{M}\) is the set of all maps that rearrange the distribution \(f\) into \(g\).
Quadratic Wasserstein distance: \(p=2\)
\[W_2^2(f,g)=\inf_{T\in\mathcal{M}}\int{}|x-T(x)|^pf(x)dx\]
Kantorovich Problem
Definition
\[\inf_\gamma\left\{\int_{X\times{}Y}c(x,y)d\gamma|\gamma\geq0,\gamma\in\Pi(\mu,\nu)\right\}\] where \(\Pi(\mu,\nu)=\{\gamma\in\mathcal{P}(X\times{}Y)|(P_X)\#\gamma=\mu,(P_Y)\#\gamma=\nu\}\), \(P_X\) and \(P_Y\) are two projections
Kantorovich Dual Problem
Consider \(\varphi\in{}L^1(\mu)\) and \(\psi\in{}L^1(\nu)\), the Kantorovich dual problem is formulated as the following: \[\sup_{\varphi,\psi}\left(\int_X\varphi{}d\mu+\int_Y\psi{}d\nu\right)\] subject to \(\varphi(x)+\psi(y)\leq{}c(x,y)\), for any \((x,y)\in{}X\times{}Y\). Note that this dual formulation is a linear optimization problem which is solvable by linear programming.
Dynamic Formulation
The Benamou-Brenier formula identifies the squared quadratic Wasserstein metric between \(\mu\) and \(\nu\) by: \[W_2^2(\mu,\nu)=\inf\int_0^1\int|v(t,x)|^2\rho(t,x)dxdt\] where infimum is taken among all the Borel fields \(v(t,x)\) that transports \(\mu\) to \(\nu\) continuously in time, satisfying the zero flux condition on the boundary: \[\begin{split}\frac{\partial\rho}{\partial{}t}+\nabla(v\rho)&=0\\\text{subject to }\rho(0,x)=d\mu,\rho(1,x)&=d\nu\end{split}\]
Monge-Ampere Equation
\[\left\{\begin{split}&det(D^2u(x))=f(x)/g(\nabla{}u(x))\\&\nabla{}u:X\rightarrow{}Y\\&u\text{ is convex}\end{split}\right.\] The optimal map is \(\nabla{}u\). Thus, the square of the quadratic Wasserstein distance has the form: \[W_2^2(f,g)=\int{}|x-\nabla{}u(x)|^2f(x)dx\]
