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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 the 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 joint optimization framework - Structured Sparse Subspace Clustering (S3C) - for learning both the affinity and the segmentation. The proposed S3C 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. Moreover, we extend the proposed S3C framework into Constrained S3C (CS3C) in which available partial side-information is incorporated into the stage of learning the affinity. We show that both the structured sparse representation and the segmentation can be found via a combination of an alternating direction method of multipliers with spectral clustering. Experiments on a synthetic data set, the Extended Yale B face data set, the Hopkins 155 motion segmentation database, and three cancer data sets demonstrate the effectiveness of our approach.

Structured Sparse Subspace Clustering: A Joint Affinity Learning and Subspace Clustering Framework