Observational data usually comes with a multimodal nature, which means that it can be naturally represented by a multi-layer graph whose layers share the same set of vertices (objects) with different edges (pairwise relationships). In this paper, we address the problem of combining different layers of the multi-layer graph for an improved clustering of the vertices compared to using layers independently. We propose two novel methods, which are based on a joint matrix factorization and a graph regularization framework respectively, to efficiently combine the spectrum of the multiple graph layers, namely the eigenvectors of the graph Laplacian matrices. In each case, the resulting combination, which we call a “joint spectrum” of multiple layers, is used for clustering the vertices. We evaluate our approaches by experiments with several real world social network datasets. Results demonstrate the superior or competitive performance of the proposed methods compared to state-of-the-art techniques and common baseline methods, such as co-regularization and summation of information from individual graphs.
 We propose a novel framework for constrained spectral clustering with pairwise constraints which specify whether two objects belong to the same cluster or not. Unlike previous methods that modify the similarity matrix with pairwise constraints, we adapt the spectral embedding towards an ideal embedding as consistent with the pairwise constraints as possible. Our formulation leads to a small semidefinite program whose complexity is independent of the number of objects in the data set and the number of pairwise constraints, making it scalable to large-scale problems. The proposed approach is applicable directly to multi-class problems, handles both must-link and cannot-link constraints, and can effectively propagate pairwise constraints. Extensive experiments on real image data and UCI data have demonstrated the efficacy of our algorithm. Clustering performance can often be greatly improved by leveraging side information. In this paper, we consider constrained clustering with pairwise constraints, which specify some pairs of objects from the same cluster or not. The main idea is to design a kernel to respect both the proximity structure of the data and the given pairwise constraints. We propose a spectral kernel learning framework and formulate it as a convex quadratic program, which can be optimally solved efficiently. Our framework enjoys several desirable features: 1) it is applicable to multi-class problems; 2) it can handle both must-link and cannot-link constraints; 3) it can propagate pairwise constraints effectively; 4) it is scalable to large-scale problems; and 5) it can handle weighted pairwise constraints. Extensive experiments have demonstrated the superiority of the proposed approach. Pairwise constraints specify whether or not two samples should be in one cluster. Although it has been successful to incorporate them into traditional clustering methods, such as K-means, little progress has been made in combining them with spectral clustering. The major challenge in designing an effective constrained spectral clustering is a sensible combination of the scarce pairwise constraints with the original affinity matrix. We propose to combine the two sources of affinity by propagating the pairwise constraints information over the original affinity matrix. Our method has a Gaussian process interpretation and results in a closed-form expression for the new affinity matrix. Experiments show it outperforms state-of-the-art constrained clustering methods in getting good clusterings with fewer constraints, and yields good image segmentation with user-specified pairwise constraints. Constrained clustering has been well-studied for algorithms like K-means and hierarchical agglomerative clustering. However, how to encode constraints into spectral clustering remains a developing area. In this paper, we propose a flexible and generalized framework for constrained spectral clustering. In contrast to some previous efforts that implicitly encode Must-Link and Cannot-Link constraints by modifying the graph Laplacian or the resultant eigenspace, we present a more natural and principled formulation, which preserves the original graph Laplacian and explicitly encodes the constraints. Our method offers several practical advantages: it can encode the degree of belief (weight) in Must-Link and Cannot-Link constraints; it guarantees to lower-bound how well the given constraints are satisfied using a user-specified threshold; and it can be solved deterministically in polynomial time through generalized eigendecomposition. Furthermore, by inheriting the objective function from spectral clustering and explicitly encoding the constraints, much of the existing analysis of spectral clustering techniques is still valid. Consequently our work can be posed as a natural extension to unconstrained spectral clustering and be interpreted as finding the normalized min-cut of a labeled graph. We validate the effectiveness of our approach by empirical results on real-world data sets, with applications to constrained image segmentation and clustering benchmark data sets with both binary and degree-of-belief constraints. Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues of symmetrical graphs Eigenvalues of subgraphs with boundary conditions Harnack inequalities Heat kernels Sobolev inequalities Advanced techniques for random walks on graphs Bibliography Index.