  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 on the Reuters dataset. Our implementation is
publicly available at https://github.com/kstant0725/SpectralNet .
