Perturbation Based Sparse Subspace Clustering

Many application areas including signal and image processing, computer vision, radar and remote sensing, bioinformatics deal with high dimensional data of various types. In these applications, the high dimensional data is not generally distributed over the whole signal space; rather it lives in the union of low dimensional subspaces. Hence, classical clustering techniques depending data distributions in centroids are not successful, and techniques that facilities the low dimensional subspace structure of big data are required. Sparse subspace clustering (SSC) technique that relies on the self-expressiveness of the data is shown to provably handle the data under noiseless case for independent and disjoint subspaces. Self-expressiveness means that each data point in a union of subspaces can be efficiently represented as a linear or affine combination of data points in the set. SSC implementation involves solving an L1 minimization problem for each data point in the space and applying spectral clustering to the affinity matrix constructed by the obtained coefficients. Despite good properties, SSC suffers from high computational complexity increasing with data point numbers. In addition, for noisy data self-expressiveness does not apply anymore. This paper proposes to use perturbed orthogonal matching pursuit (POMP) within SSC framework for robust and computationally efficient estimation of the number of subspaces, their dimensions, and the segmentation of the data into each subspace. POMP was shown to be successful in recovering sparse signals under random basis perturbations, which is actually the case in corrupted data clustering. Our initial results for simulated clustering datasets show that the proposed POMP-SSC technique provides both computational efficiency and high clustering performance compared to classical SSC implementation.