Scaling analysis of affinity propagation

We analyze and exploit some scaling properties of the affinity propagation (AP) clustering algorithm proposed by Frey and Dueck [Science 315, 972 (2007)]. Following a divide and conquer strategy we setup an exact renormalization-based approach to address the question of clustering consistency, in particular, how many cluster are present in a given data set. We first observe that the divide and conquer strategy, used on a large data set hierarchically reduces the complexity O (N2) 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 such as 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, as a result of an exact decimation procedure. From this observation, a strategy based on AP can be defined to find out how many clusters are present in a given data set.