This paper presents a concise tutorial on spectral clustering for broad spectrum graphs which include unipartite (undirected) graph, bipartite graph, and directed graph. We show how to transform bipartite graph and directed graph into corresponding unipartite graph, therefore allowing a unified treatment to all cases. In bipartite graph, we show that the relaxed solution to the @math -way co-clustering can be found by computing the left and right eigenvectors of the data matrix. This gives a theoretical basis for @math -way spectral co-clustering algorithms proposed in the literatures. We also show that solving row and column co-clustering is equivalent to solving row and column clustering separately, thus giving a theoretical support for the claim: column clustering implies row clustering and vice versa''. And in the last part, we generalize the Ky Fan theorem---which is the central theorem for explaining spectral clustering---to rectangular complex matrix motivated by the results from bipartite graph analysis.
Principal component analysis (PCA) is a widely used statistical technique for unsupervised dimension reduction. K-means clustering is a commonly used data clustering for performing unsupervised learning tasks. Here we prove that principal components are the continuous solutions to the discrete cluster membership indicators for K-means clustering. New lower bounds for K-means objective function are derived, which is the total variance minus the eigenvalues of the data covariance matrix. These results indicate that unsupervised dimension reduction is closely related to unsupervised learning. Several implications are discussed. On dimension reduction, the result provides new insights to the observed effectiveness of PCA-based data reductions, beyond the conventional noise-reduction explanation that PCA, via singular value decomposition, provides the best low-dimensional linear approximation of the data. On learning, the result suggests effective techniques for K-means data clustering. DNA gene expression and Internet newsgroups are analyzed to illustrate our results. Experiments indicate that the new bounds are within 0.5-1.5 of the optimal values. 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.