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Matching distance

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Title: Matching distance  
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Subject: Size theory
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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 \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.


See also

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