Spectral Relaxation for K-means Clustering

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 nd 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.

• Type: Conference
• Year: 2001
• Where: NIPS
• Authors: Hongyuan Zha, Xiaofeng He, Chris H. Q. Ding, Ming Gu, Horst D. Simon
• Keywords: Gram Matrix, NIPS 2001, NIPS 2007, Sum-of-squares Cost Function, Trace Maximization Problem