Matching distance

In mathematics, the matching distance  is a metric on the space of size functions.



The core of the definition of matching distance is the observation that the information contained in a size function can be combinatorially stored in a formal series of lines and points of the plane, called respectively cornerlines and cornerpoints.

Given two size functions $$\ell_1$$ and $$\ell_2$$, let  $$C_1$$ (resp. $$C_2$$) be the multiset of all cornerpoints and cornerlines for $$\ell_1$$ (resp. $$\ell_2$$) counted with their multiplicities, augmented by adding a countable infinity of points of the diagonal $$\{(x,y)\in \R^2: x=y\}$$.

The matching distance between $$\ell_1$$ and $$\ell_2$$ is given by $$d_\text{match}(\ell_1, \ell_2)=\min_\sigma\max_{p\in C_1}\delta (p,\sigma(p))$$ where $$\sigma$$ varies among all the bijections between $$C_1$$ and $$C_2$$ and


 * $$\delta\left((x,y),(x',y')\right)=\min\left\{\max \{|x-x'|,|y-y'|\}, \max\left\{\frac{y-x}{2},\frac{y'-x'}{2}\right\}\right\}.$$

Roughly speaking, the matching distance $$d_\text{match}$$ between two size functions is the minimum, over all the matchings between the cornerpoints of the two size functions, of the maximum of the $$L_\infty$$-distances between two matched cornerpoints. Since two size functions can have a different number of cornerpoints, these can be also matched to points of the diagonal $$\Delta$$. Moreover, the definition of $$\delta$$ implies that matching two points of the diagonal has no cost.