Scaling Analysis of Affinity Propagation
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We analyze and exploit some scaling properties of the Affinity Propagation (AP) clustering algorithm proposed by Frey and Dueck (2007). First we observe that a divide and conquer strategy, used on a large data set hierarchically reduces the complexity O(N^2) to O(N^(h+2)/(h+1)), for a data-set of size N and a depth h of the hierarchical strategy. For a data-set embedded in a d-dimensional space, we show that this is obtained without notably damaging the precision except in dimension d=2. In fact, for d larger than 2 the relative loss in precision scales like N^(2-d)/(h+1)d. Finally, under some conditions we observe that there is a value s^* of the penalty coefficient, a free parameter used to fix the number of clusters, which separates a fragmentation phase (for s<s^*) from a coalescent one (for s>s^*) of the underlying hidden cluster structure. At this precise point holds a self-similarity property which can be exploited by the hierarchical strategy to actually locate its position. From this observation, a strategy based on can be defined to find out how many clusters are present in a given dataset.
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