Fast greedy algorithm for subspace clustering from corrupted and incomplete data

We describe the Fast Greedy Sparse Subspace Clustering (FGSSC) algorithm providing an efficient method for clustering data belonging to a few low-dimensional linear or affine subspaces. The main difference of our algorithm from predecessors is its ability to work with noisy data having a high rate of erasures (missed entries with the known coordinates) and errors (corrupted entries with unknown coordinates). The greedy approach consists in usage of a basic algorithm on data out of the domain of the basic algorithm reliability. When the algorithm fails and its output looses some desirable features of the solution, usually it still brings some information about ”the largest features” of the solution. The greedy algorithm extracts those features and uses them in an iterative way by launching the basic algorithm with the additional information mined out on the previous iterations. Such scheme requires an additional time consuming algorithm loop. We discuss here how to implement the fast version of the greedy algorithm with the maximum efficiency whose greedy strategy is incorporated into iterations of the basic algorithm. We provide numerical evidences that, in the subspace clustering capability, the fast greedy algorithm outperforms not only the existing state-of-the art SSC algorithm taken by the authors as a basic algorithm but also the recent GSSC algorithm. At the same time, its computational cost is only slightly higher than the cost of SSC. The numerical evidence of the algorithm significant advantage is presented for a few synthetic models as well as for the Extended Yale B dataset of facial images. In particular, the face recognition misclassification rate turned out to be 6–20 times lower than for the SSC algorithm. We provide also the numerical evidence that the FGSSC algorithm is able to perform clustering of corrupted data efficiently even when the sum of subspace dimensions significantly exceeds the dimension of the ambient space.