The cliques considered here are defined by circles (in two dimensions) or spheres (in three dimensions.) In the most basic version, we have one clique for each cluster, and the clique is defined as the smallest circle containing a pre-specified proportion p of the points from the cluster in question. If the clusters are well separated, we can even use p = 1. We define the density of a clique as the number of points per unit area. In general, we want to build cliques with high density.
Ideally, we want each cluster in the training set to be covered by a small number of (possibly slightly overlapping) cliques, each one having a high density. Also, as a general rule, a training set point can only belong to one clique, and (ideally) to only one cluster. But the circles associated with two cliques are allowed to overlap.
It’s an interesting approach, and I can see how it’d be faster than K Nearest Neighbors, but I do wonder how accurate the results would be in comparison to KNN.