Matching distance

In mathematics, the matching distance^{[1] [2]} 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 ${\displaystyle \ell _{1}}${\displaystyle \ell _{1}} and ${\displaystyle \ell _{2}}${\displaystyle \ell _{2}}, let ${\displaystyle C_{1}}${\displaystyle C_{1}} (resp. ${\displaystyle C_{2}}${\displaystyle C_{2}}) be the multiset of all cornerpoints and cornerlines for ${\displaystyle \ell _{1}}${\displaystyle \ell _{1}} (resp. ${\displaystyle \ell _{2}}${\displaystyle \ell _{2}}) counted with their multiplicities, augmented by adding a countable infinity of points of the diagonal ${\displaystyle \{(x,y)\in \mathbb {R} ^{2}:x=y\}}${\displaystyle \{(x,y)\in \mathbb {R} ^{2}:x=y\}}.

The matching distance between ${\displaystyle \ell _{1}}${\displaystyle \ell _{1}} and ${\displaystyle \ell _{2}}${\displaystyle \ell _{2}} is given by ${\displaystyle d_{\text{match}}(\ell _{1},\ell _{2})=\min _{\sigma }\max _{p\in C_{1}}\delta (p,\sigma (p))}${\displaystyle d_{\text{match}}(\ell _{1},\ell _{2})=\min _{\sigma }\max _{p\in C_{1}}\delta (p,\sigma (p))} where ${\displaystyle \sigma }${\displaystyle \sigma } varies among all the bijections between ${\displaystyle C_{1}}${\displaystyle C_{1}} and ${\displaystyle C_{2}}${\displaystyle C_{2}} and

${\displaystyle \delta \left((x,y),(x',y')\right)=\min \left\{\max\{|x-x'|,|y-y'|\},\max \left\{{\frac {y-x}{2}},{\frac {y'-x'}{2}}\right\}\right\}.}${\displaystyle \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 ${\displaystyle d_{\text{match}}}${\displaystyle 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 ${\displaystyle L_{\infty }}${\displaystyle 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 ${\displaystyle \Delta }${\displaystyle \Delta }. Moreover, the definition of ${\displaystyle \delta }${\displaystyle \delta } implies that matching two points of the diagonal has no cost.

• Size theory
• Size function
• Size functor
• Size homotopy group
• Natural pseudodistance

• Topology