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In this paper, we revisit the popular affinity matrix based on the anchor graph and point out that the spectral embedding obtained using symmetric normalized Laplacian is only a side view of the bipartite structure. Based on the analysis, we propose Fast Spectral Clustering based on the Random Walk Laplacian (FRWL) method to explicitly balance the popularity of anchors and the independence of data points, which is especially important for clustering of boundary points.

We present a novel method to hierarchically cluster networked data allowing nodes to simultaneously belong to multiple clusters. Given a network, our method outputs a cut metric on the underlying node set, which can be related to data coverings at different resolutions. The cut metric is obtained by averaging a set of ultrametrics, which are themselves the output of (non-overlapping) hierarchically clustering noisy versions of the original network of interest. The resulting algorithm is illustrated in synthetic networks and is used to classify handwritten digits from the MNIST database.