Monge-Elkan similarity, generalized

$$sim_{GME}(x, y) = \left(\frac{1}{|x|} \sum_{i=1}^{|x|} (\max_{j=1,|y|}(sim(x_i, y_j)))^m\right)^{\frac{1}{m}}$$

Where $sim$ can be any other similarity measure.

Extension of Monge-Elkan similarity.

Reading

• Jiménez, Sergio, Claudia Jeanneth Becerra, Alexander Gelbukh and Fabio A. González. “Generalized Mongue-Elkan Method for Approximate Text String Comparison.” CICLing (2009).

Generalized mean (Hölder mean)

$$x(a) = \left( \frac{1}{n} \sum_{i=1}^{n} a_i^m \right)^{1/m}$$

• $m = 1$ arithmetic mean
• $m = 2$ quadratic mean
• $m \rightarrow 0$ geometric mean
• $m \rightarrow -1$ harmonic mean
• $m \rightarrow \infty$ maximum
• $m \rightarrow -\infty$ minimum