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Downsampling for sparse subspace clustering

Sparse subspace clustering (SSC) is a technique to partition unlabeled samples according to the subspaces they locate in. With the rapid increase of data amount, efficiently downsampling a big dataset, while at the same time keeping the structure of subspaces, becomes an important topic for SSC. In order to reduce the computational cost while preserving clustering accuracy, a new approach of SSC with downsampling (SSCD) is proposed in this paper. In SSCD, the numbers of samples located in respective subspaces are estimated utilizing the ℓ1 norm of the sparse representation. Then a downsampling strategy is designed to decimate samples with the probabilities that are in reverse ratio to the amounts of samples in respective subspaces. As a consequence, the samples in different subspaces are expected to be balanced after the downsampling operation. Theoretical analysis proves the correctness of the proposed strategy. Numerical simulations also verify the efficiency of SSCD.