The Basic CLICK Algorithm

The CLICK algorithm represents the input data as a weighted similarity graph G = (V,E). In this graph vertices correspond to elements and edge weights are derived from the similarity values. The weight w_ij of an edge (i,j) reflects the probability that i and j are mates, and is set to be

$$w_{ij}=\log{\frac{p_{mates}f(S_{ij}\vert\textrm{$i$ ,$j$\spac... ...{mates})f(S_{ij}\vert\textrm{$i$ ,$j$\space are non-mates})}}$$

where $f(S_{ij}\vert\textrm{$i$ ,$j$\space are mates})=f(S_{ij}\vert\mu_T,\sigma_T)$ is the value of the probability density function for mates at S_ij:

$$f(S_{ij}\vert\textrm{$i$ ,$j$\space are mates})=\frac{1}{\sqrt{2\pi}\sigma_T} e^{-\frac{(S_{ij}-\mu_T)^2}{2\sigma_T^2}}$$

Similarly, $f(S_{ij}\vert\textrm{$i$ ,$j$\space are non-mates})$ is the value of the probability density function for non-mates. The basic CLICK algorithm is defined in figure 11.13.

  

Figure 11.13: The Basic-CLICK algorithm

The idea behind the algorithm the following: given a connected graph G, we would like to decide whether V(G) is a subset of some true cluster, or V(G) contains elements from at least two true clusters. In the first case we say that G is pure. In order to make this decision we test for each cut C in G the following two hypotheses:

• H₀^C: C contains only edges between non-mates.
• H₁^C: C contains only edges between mates.

G is declared a kernel if H₁ is more probable for all cuts. Using the following lemma (11.6), we can simply calculate the minimum weighted cut to determine whether G is a kernel.