## Abstract

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.

Original language | English (US) |
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Article number | 066102 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 81 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2010 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics