  Spectral Clustering as a relaxation of the normalized/ratio cut has become
one of the standard graph-based clustering methods. Existing methods for the
computation of multiple clusters, corresponding to a balanced $k$-cut of the
graph, are either based on greedy techniques or heuristics which have weak
connection to the original motivation of minimizing the normalized cut. In this
paper we propose a new tight continuous relaxation for any balanced $k$-cut
problem and show that a related recently proposed relaxation is in most cases
loose leading to poor performance in practice. For the optimization of our
tight continuous relaxation we propose a new algorithm for the difficult
sum-of-ratios minimization problem which achieves monotonic descent. Extensive
comparisons show that our method outperforms all existing approaches for ratio
cut and other balanced $k$-cut criteria.
