Multiclass Spectral Clustering

We propose a principled account on multiclass spectral clustering. Given a discrete clustering formulation, we first solve a relaxed continuous optimization problem by eigendecomposition. We clarify the role of eigenvectors as a generator of all optimal solutions through orthonormal transforms. We then solve an optimal discretization problem, which seeks a discrete solution closest to the continuous optima. The discretization is efficiently computed in an iterative fashion using singular value decomposition and nonmaximum suppression. The resulting discrete solutions are nearly global-optimal. Our method is robust to random initialization and converges faster than other clustering methods. Experiments on real image segmentation are reported. Spectral graph partitioning methods have been successfully applied to circuit layout [3, 1], load balancing [4] and image segmentation [10, 6]. As a discriminative approach, they do not make assumptions about the global structure of data. Instead, local evidence on how likely two data points belong to the same class is first collected and a global decision is then made to divide all data points into disjunct sets according to some criterion. Often, such a criterion can be interpreted in an embedding framework, where the grouping relationships among data points are preserved as much as possible in a lower-dimensional representation. What makes spectral methods appealing is that their global-optima in the relaxed continuous domain are obtained by eigendecomposition. However, to get a discrete solution from eigenvectors often requires solving another clustering problem, albeit in a lower-dimensional space. That is, eigenvectors are treated as geometrical coordinates of a point set. Various clustering heuristics such as Kmeans [10, 9], transportation [2], dynamic programming [1], greedy pruning or exhaustive search [3, 10] are subsequently employed on the new point set to retrieve partitions. We show that there is a principled way to recover a discrete optimum. This is based on a fact that the continuous optima consist not only of the eigenvectors, but of a whole family spanned by the eigenvectors through orthonormal transforms. The goal is to find the right orthonormal transform that leads to a discretization.