Grassmannian Manifold Optimization Assisted Sparse Spectral Clustering

Abstract

Spectral Clustering is one of pioneered clustering methods in machine learning and pattern recognition field. It relies on the spectral decomposition criterion to learn a low-dimensonal embedding of data for a basic clustering algorithm such as the k-means. The recent sparse Spectral clustering (SSC) introduces the sparsity for the similarity in low-dimensional space by enforcing a sparsity-induced penalty, resulting a non-convex optimization, and the solution is calculated through a relaxed convex problem via the standard ADMM (Alternative Direction Method of Multipliers), rather than inferring latent representation from eigen-structure. This paper provides a direct solution as solving a new Grassmann optimization problem. By this way calculating latent embedding becomes part of optmization on manifolds and the recently developed manifold optimization methods can be applied. It turns out the learned new features are not only very informative for clustering, but also more intuitive and effective in visualization after dimensionality reduction. We conduct empirical studies on simulated datasets and several real-world benchmark datasets to validate the proposed methods. Experimental results exhibit the effectiveness of this new manifold-based clustering and dimensionality reduction method.