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 \(g\).

Convert LP to a Standard Form

\[\min{}c^\top{}x,\text{subject to }Ax=b\land{}x\geq0\]

For Inequality

• \(x+y\geq{}a\rightarrow{}x+y-z=a,z\geq0\)
• \(x+y\leq{}a\rightarrow{}x+y+s=a,s\geq0\)
• Unrestricted \(\rightarrow{}x=y-z,y\geq0,z\geq0\)