Exponentially Consistent Nonparametric Clustering of Data Streams

In this paper, we consider nonparametric clustering of $M$ independent and identically distributed (i.i.d.) data streams generated from unknown distributions. The distributions of the $M$ data streams belong to $K$ underlying distribution clusters. Existing results on exponentially consistent nonparametric clustering algorithms, like single linkage-based (SLINK) clustering and $k$-medoids distribution clustering, assume that the maximum intra-cluster distance ($d_L$) is smaller than the minimum inter-cluster distance ($d_H$). First, in the fixed sample size (FSS) setting, we show that exponential consistency can be achieved for SLINK clustering under a less strict assumption, $d_I < d_H$, where $d_I$ is the maximum distance between any two sub-clusters of a cluster that partition the cluster. Note that $d_I < d_L$ in general. Our results show that SLINK is exponentially consistent for a larger class of problems than $k$-medoids distribution clustering. We also identify examples where $k$-medoids clustering is unable to find the true clusters, but SLINK is exponentially consistent. Then, we propose a sequential clustering algorithm, named SLINK-SEQ, based on SLINK and prove that it is also exponentially consistent. Simulation results show that the SLINK-SEQ algorithm requires fewer expected number of samples than the FSS SLINK algorithm for the same probability of error.