Nonnegative Lagrangian Relaxation of K-Means and Spectral Clustering

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.