Metrics and Similarity Measures

Metric Space (X,d) $ d:X \times X \rightarrow \Re ^{+} $

X is set, not necessarily vector space

$ x, y, z \in X $

1. $ d(x,y)=d(y,x) $
2. $ d(x,z)\leq d(x,y)+d(y,z) $
3. $ d(x,y) \geq 0, d(x,y)=0 \Leftrightarrow x=y) $

If X is vector space, metric can be induced by the norm $ ||\cdot|| $.

$ d(x,y)=||y-x|| $

Norm is defined as follows

$ ||\cdot||: X \rightarrow \Re ^{+} $

1. $ |x| \geq 0, ||x||=0 \Leftrightarrow x=0 $
2. $ ||\alpha x||=|\alpha | ||x|| $
3. $ ||x+y|| \leq ||x|| + ||y|| $

Defining metric, we can measure similarity of elements of set X.

Example of metric

1. Minkowski Metric $ \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p} $
2. Riemannian Metric $ D(\vec{x_1},\vec{x_2})=\sqrt{(\vec{x_1}-\vec{x_2})^\top \mathbb{M}(\vec{x_1}-\vec{x_2})} $
3. Tanimoto metric $ D(S_1, S_2) = \frac {|S_1|+|S_2|-2|S_1 \bigcap S_2| }{|S_1|+|S_2|-|S_1 \bigcap S_2|} $
4. Procrustes metric $ D(p,\bar p)= min_{R,T} \sum_{i=1}^n {\begin{Vmatrix} Rp_i+T-\bar p_i \end{Vmatrix}} _{L^2} $, R: Rotation, T: Translation

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