### Abstract

In this paper we present a novel method for unraveling the hierarchical clusters in a given dataset using the Gershgorin circle theorem. The Gershgorin circle theorem provides upper bounds on the eigenvalues of the normalized Laplacian matrix. This can be utilized to determine the ideal range for the number of clusters (k) at different levels of hierarchy in a given dataset. The obtained intervals help to reduce the search space for identifying the ideal value of k at each level. Another advantage is that we don't need to perform the computationally expensive eigen-decomposition step to obtain the eigenvalues and eigenvectors. The intervals provided for k can be considered as input for any spectral clustering method which uses a normalized Laplacian matrix. We show the effectiveness of the method in combination with a spectral clustering method to generate hierarchical clusters for several synthetic and real world datasets.

Original language | English |
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Pages (from-to) | 1-7 |

Number of pages | 7 |

Journal | Pattern Recognition Letters |

Volume | 55 |

DOIs | |

Publication status | Published - 1 Apr 2015 |

Externally published | Yes |

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### Keywords

- Eigengap
- Gershgorin circle theorem
- k clusters

### ASJC Scopus subject areas

- Software
- Signal Processing
- Computer Vision and Pattern Recognition
- Artificial Intelligence

### Cite this

*Pattern Recognition Letters*,

*55*, 1-7. https://doi.org/10.1016/j.patrec.2014.12.007