Exponentially Consistent Nonparametric Linkage-Based Clustering of Data Sequences
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| Format: | Preprint |
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2024
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| author | Singh, Bhupender Rajagopalan, Ananth Ram Bhashyam, Srikrishna |
| author_facet | Singh, Bhupender Rajagopalan, Ananth Ram Bhashyam, Srikrishna |
| contents | In this paper, we consider nonparametric clustering of $M$ independent and identically distributed (i.i.d.) data sequences generated from {\em unknown} distributions. The distributions of the $M$ data sequences 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. Thus, our results show that SLINK is exponentially consistent for a larger class of problems than previously known. In our simulations, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_13922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponentially Consistent Nonparametric Linkage-Based Clustering of Data Sequences Singh, Bhupender Rajagopalan, Ananth Ram Bhashyam, Srikrishna Machine Learning Information Theory Signal Processing In this paper, we consider nonparametric clustering of $M$ independent and identically distributed (i.i.d.) data sequences generated from {\em unknown} distributions. The distributions of the $M$ data sequences 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. Thus, our results show that SLINK is exponentially consistent for a larger class of problems than previously known. In our simulations, 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. |
| title | Exponentially Consistent Nonparametric Linkage-Based Clustering of Data Sequences |
| topic | Machine Learning Information Theory Signal Processing |
| url | https://arxiv.org/abs/2411.13922 |