Spectral Relaxation for K-means Clustering (2001)
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BibTeX
@INPROCEEDINGS{Zha01spectralrelaxation,
author = {Hongyuan Zha and Xiaofeng He and Chris Ding and Horst Simon and Ming Gu},
title = {Spectral Relaxation for K-means Clustering},
booktitle = {},
year = {2001},
pages = {1057--1064},
publisher = {MIT Press}
}
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Abstract
The popular K-means clustering partitions a data set by minimizing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function.
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