Nonnegative Lagrangian relaxation of k-means and spectral clustering

Authors:
Chris Ding;Xiaofeng He;Horst D. Simon
Affiliations:
Lawrence Berkeley National Laboratory, Berkeley, CA, USA.;Lawrence Berkeley National Laboratory, Berkeley, CA, USA.;Lawrence Berkeley National Laboratory, Berkeley, CA, USA.
Venue:
ECML'05 Proceedings of the 16th European conference on Machine Learning
Year:
2005

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Abstract

We show that K-means and spectral clustering objective functions can be written as a trace of quadratic forms. Instead of relaxation by eigenvectors, we propose a novel relaxation maintaining the nonnegativity of the cluster indicators and thus give the cluster posterior probabilities, therefore resolving cluster assignment difficulty in spectral relaxation. We derive a multiplicative updating algorithm to solve the nonnegative relaxation problem. The method is briefly extended to semi-supervised classification and semi-supervised clustering.