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.
In addition to the general graph-based data processing, there is a unique branch in graph theory that is devoted to analyzing the spectrum of the graphs, the spectral graph theory. The manuscript by Chung @cite_25 gives a good introduction to this field. Among various methods that are developed, we particularly emphasize the so-called spectral clustering algorithm, which has become one of the major graph-based clustering techniques. Due to its promising performance and close links to other well-studied mathematical fields, a large number of variants of the original algorithm has been proposed, such as the constrained spectral clustering algorithm @cite_8 @cite_24 @cite_42 @cite_32 @cite_2 . In general, these works have suggested different ways to incorporate constraints in the clustering task. Among them, @cite_42 has proposed a regularization framework in the graph spectral domain, which provides the closest methodology to our work.
@cite_8 @cite_42 @cite_32 @cite_24 @cite_2 @cite_25
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Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix @math , one may look at the spectrum of @math for a properly chosen @math . The issue is that the spectrum of @math might be contaminated by non-informational top eigenvalues, e.g., due to scale variations in the data, and the application of @math aims to remove these. Designing a good functional @math (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, @math where @math denotes the adjacency matrix and @math is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance @math , and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd o s-R 'enyi ensemble, which has no spectral gap, it is shown that graph powering produces a maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to massoulie-STOC , settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts massoulie-STOC,Mossel_SBM2,bordenave,colin3 . This is illustrated on a geometric block model that is dense in cliques. Based on spectral graph theory, spectral clustering is an optimal graph partition problem. It has been proven that the spectral clustering is equivalent to nonnegative matrix factorization (NMF) under certain conditions. Based on the equivalence, some spectral clustering methods are proposed, but the global discriminative information of the dataset is neglected. In this paper, based on the equivalence between spectral clustering and NMF, we simultaneously maximize the between-class scatter matrix and minimize the within-class scatter matrix to enhance the discriminating power. We integrate the geometrical structure and discriminative structure in a joint framework. With a global discriminative regularization term added into the nonnegative matrix factorization framework, we propose two novel spectral clustering methods, named global discriminative-based nonnegative and spectral clustering (GDBNSC-Ncut and GDBNSC-Rcut) These new spectral clustering algorithms can preserve both the global geometrical structure and global discriminative structure. The intrinsic geometrical information of the dataset is detected, and clustering quality is improved with enhanced discriminating power. In addition, the proposed algorithms also have very good abilities of handling out-of-sample data. Experimental results on real word data demonstrate that the proposed algorithms outperform some state-of-the-art methods with good clustering qualities. We maximize the between-class scatter matrix.Meanwhile, we minimize the within-class scatter matrix.We integrate the geometrical structure and discriminative structure in a framework.A global discriminative regularization term added into the objective functions.The new algorithms can preserve the global geometrical and discriminative structure. Typical graph-theoretic approaches for semi-supervised classification infer labels of unlabeled instances with the help of graph Laplacians. Founded on the spectral decomposition of the graph Laplacian, this paper learns a kernel matrix via minimizing the leave-one-out classification error on the labeled instances. To this end, an efficient algorithm is presented based on linear programming, resulting in a transductive spectral kernel. The idea of our algorithm stems from regularization methodology and also has a nice interpretation in terms of spectral clustering. A simple classifier can be readily built upon the learned kernel, which suffices to give prediction for any data point aside from those in the available dataset. Besides this usage, the spectral kernel can be effectively used in tandem with conventional kernel machines such as SVMs. We demonstrate the efficacy of the proposed algorithm through experiments carried out on challenging classification tasks. Spectral clustering is a popular and successful approach for partitioning the nodes of a graph into clusters for which the ratio of outside connections compared to the volume (sum of degrees) is small. In order to partition into k clusters, one first computes an approximation of the bottom k eigenvectors of the (normalized) Laplacian of G, uses it to embed the vertices of G into k-dimensional Euclidean space R^k, and then partitions the resulting points via a k-means clustering algorithm. It is an important task for theory to explain the success of spectral clustering. (COLT, 2015) made an important step in this direction. They showed that spectral clustering provably works if the gap between the (k+1)-th and the k-th eigenvalue of the normalized Laplacian is sufficiently large. They proved a structural and an algorithmic result. The algorithmic result needs a considerably stronger gap assumption and does not analyze the standard spectral clustering paradigm; it replaces spectral embedding by heat kernel embedding and k-means clustering by locality sensitive hashing. We extend their work in two directions. Structurally, we improve the quality guarantee for spectral clustering by a factor of k and simultaneously weaken the gap assumption. Algorithmically, we show that the standard paradigm for spectral clustering works. Moreover, it even works with the same gap assumption as required for the structural result.