A basic problem in spectral clustering is the following. If a solution obtained from the spectral relaxation is close to an integral solution, is it possible to find this integral solution even though they might be in completely different basis? In this paper, we propose a new spectral clustering algorithm. It can recover a @math -partition such that the subspace corresponding to the span of its indicator vectors is @math close to the original subspace in spectral norm with @math being the minimum possible ( @math always). Moreover our algorithm does not impose any restriction on the cluster sizes. Previously, no algorithm was known which could find a @math -partition closer than @math . We present two applications for our algorithm. First one finds a disjoint union of bounded degree expanders which approximate a given graph in spectral norm. The second one is for approximating the sparsest @math -partition in a graph where each cluster have expansion at most @math provided @math where @math is the @math eigenvalue of Laplacian matrix. This significantly improves upon the previous algorithms, which required @math .
Spectral methods have been successfully used for clustering tasks @cite_2 arising in many different areas such as VLSI @cite_9 , machine learning, data analysis @cite_1 and computer vision @cite_17 @cite_14 . They are usually obtained by formulating the clustering task as a combinatorial optimization problem (such as sparsest normalized cuts @cite_17 ), then solving the corresponding basic SDP relaxation, whose solution is often given by @math extremal eigenvectors of an associated matrix.
@cite_14 @cite_9 @cite_1 @cite_2 @cite_17
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The technique of spectral clustering is widely used to segment a range of data from graphs to images. Our work marks a natural progression of spectral clustering from the original passive unsupervised formulation to our active semi-supervised formulation. We follow the widely used area of constrained clustering and allow supervision in the form of pair wise relations between two nodes: Must-Link and Cannot-Link. Unlike most previous constrained clustering work, our constraints are specified incrementally by querying an oracle (domain expert). Since in practice, each query comes with a cost, our goal is to maximally improve the result with as few queries as possible. The advantages of our approach include: 1) it is principled by querying the constraints which maximally reduce the expected error, 2) it can incorporate both hard and soft constraints which are prevalent in practice. We empirically show that our method significantly outperforms the baseline approach, namely constrained spectral clustering with randomly selected constraints, on UCI benchmark data sets. 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. 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. Constrained clustering has been well-studied for algorithms such as K-means and hierarchical clustering. However, how to satisfy many constraints in these algorithmic settings has been shown to be intractable. One alternative to encode many constraints is to use spectral clustering, which remains a developing area. In this paper, we propose a flexible framework for constrained spectral clustering. In contrast to some previous efforts that implicitly encode Must-Link (ML) and Cannot-Link (CL) constraints by modifying the graph Laplacian or constraining the underlying eigenspace, we present a more natural and principled formulation, which explicitly encodes the constraints as part of a constrained optimization problem. Our method offers several practical advantages: it can encode the degree of belief in ML and CL constraints; it guarantees to lower-bound how well the given constraints are satisfied using a user-specified threshold; it can be solved deterministically in polynomial time through generalized eigendecomposition. Furthermore, by inheriting the objective function from spectral clustering and encoding the constraints explicitly, much of the existing analysis of unconstrained spectral clustering techniques remains valid for our formulation. We validate the effectiveness of our approach by empirical results on both artificial and real datasets. We also demonstrate an innovative use of encoding large number of constraints: transfer learning via constraints.