Part of this work was done when Dejiao Zhang was doing an internship at Technicolor Research. Both Dejiao Zhang and Laura Balzano’s participations were funded by DARPA-16-43-D3M-FP-037. Both Yifan Sun and Brian Eriksson's participation occurred while also at Technicolor Research.
Recent work @cite_2 extends spectral clustering by replacing the eigenvector representation of data with the embeddings from a deep autoencoder. Since training an autoencoder is linear in the number of samples @math , finding the embeddings is much more scalable than traditional spectral clustering, despite the difficulties in training autoencoders. However, this approach requires a @math normalized adjacency matrix as input, which is a heavy burden on both computation and memory for very large @math .
@cite_2
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Spectral clustering is a leading and popular technique in unsupervised data analysis. Two of its major limitations are scalability and generalization of the spectral embedding (i.e., out-of-sample-extension). In this paper we introduce a deep learning approach to spectral clustering that overcomes the above shortcomings. Our network, which we call SpectralNet, learns a map that embeds input data points into the eigenspace of their associated graph Laplacian matrix and subsequently clusters them. We train SpectralNet using a procedure that involves constrained stochastic optimization. Stochastic optimization allows it to scale to large datasets, while the constraints, which are implemented using a special purpose output layer, allow us to keep the network output orthogonal. Moreover, the map learned by SpectralNet naturally generalizes the spectral embedding to unseen data points. To further improve the quality of the clustering, we replace the standard pairwise Gaussian affinities with affinities leaned from unlabeled data using a Siamese network. Additional improvement can be achieved by applying the network to code representations produced, e.g., by standard autoencoders. Our end-to-end learning procedure is fully unsupervised. In addition, we apply VC dimension theory to derive a lower bound on the size of SpectralNet. State-of-the-art clustering results are reported for both the MNIST and Reuters datasets. Recently deep learning has been successfully adopted in many applications such as speech recognition and image classification. In this work, we explore the possibility of employing deep learning in graph clustering. We propose a simple method, which first learns a nonlinear embedding of the original graph by stacked autoencoder, and then runs k-means algorithm on the embedding to obtain clustering result. We show that this simple method has solid theoretical foundation, due to the similarity between autoencoder and spectral clustering in terms of what they actually optimize. Then, we demonstrate that the proposed method is more efficient and flexible than spectral clustering. First, the computational complexity of autoencoder is much lower than spectral clustering: the former can be linear to the number of nodes in a sparse graph while the latter is super quadratic due to eigenvalue decomposition. Second, when additional sparsity constraint is imposed, we can simply employ the sparse autoencoder developed in the literature of deep learning; however, it is nonstraightforward to implement a sparse spectral method. The experimental results on various graph datasets show that the proposed method significantly outperforms conventional spectral clustering, which clearly indicates the effectiveness of deep learning in graph clustering. Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the non-backtracking walk on the graph, leads to improved performance in detecting clusters, and even to optimal performance for the stochastic block model. Here, we propose to use instead a simpler object, a symmetric real matrix known as the Bethe Hessian operator, or deformed Laplacian. We show that this approach combines the performances of the non-backtracking operator, thus detecting clusters all the way down to the theoretical limit in the stochastic block model, with the computational, theoretical and memory advantages of real symmetric matrices. We introduce a novel spectral clustering algorithm that allows us to automatically determine the number of clusters in a dataset. The algorithm is based on a theoretical analysis of the spectral properties of block diagonal affinity matrices; in contrast to established methods, we do not normalise the rows of the matrix of eigenvectors, and argue that the non-normalised data contains key information that allows the automatic determination of the number of clusters present. We present several examples of datasets successfully clustered by our algorithm, both artificial and real, obtaining good results even without employing refined feature extraction techniques