Structured Sparse Subspace Clustering: A Unified Optimization Framework

Subspace clustering refers to the problem of segmenting data drawn from a union of subspaces. State of the art approaches for solving this problem follow a two-stage approach. In the first step, an affinity matrix is learned from the data using sparse or low-rank minimization techniques. In the second step, the segmentation is found by applying spectral clustering to this affinity. While this approach has led to state of the art results in many applications, it is suboptimal because it does not exploit the fact that the affinity and the segmentation depend on each other. In this paper, we propose a unified optimization framework for learning both the affinity and the segmentation. Our framework is based on expressing each data point as a structured sparse linear combination of all other data points, where the structure is induced by a norm that depends on the unknown segmentation. We show that both the segmentation and the structured sparse representation can be found via a combination of an alternating direction method of multipliers with spectral clustering. Experiments on a synthetic data set, the Hopkins 155 motion segmentation database, and the Extended Yale B data set demonstrate the effectiveness of our approach.