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Sparse subspace clustering (SSC) relies on sparse regression for accurateneighbor identification. Inspired by recent progress in compressive sensing,this paper proposes a new sparse regression scheme for SSC via two-stepreweighted $\ell_1$-minimization, which also generalizes a two-step$\ell_1$-minimization algorithm introduced by E. J. Cand\`es et al in [TheAnnals of Statistics, vol. 42, no. 2, pp. 669-699, 2014] without incurringextra algorithmic complexity. To fully exploit the prior information offered bythe computed sparse representation vector in the first step, our approachplaces a weight on each component of the regression vector, and solves aweighted LASSO in the second step. We propose a data weighting rule suitablefor enhancing neighbor identification accuracy. Then, under the formulation ofthe dual problem of weighted LASSO, we study in depth the theoretical neighborrecovery rates of the proposed scheme. Specifically, an interesting connectionbetween the locations of nonzeros of the optimal sparse solution to theweighted LASSO and the indexes of the active constraints of the dual problem isestablished. Afterwards, under the semi-random model, analytic probabilitylower/upper bounds for various neighbor recovery events are derived. Ouranalytic results confirm that, with the aid of data weighting and if the priorneighbor information is enough accurate, the proposed scheme with a higherprobability can produce many correct neighbors and few incorrect neighbors ascompared to the solution without data weighting. Computer simulations areprovided to validate our analytic study and evidence the effectiveness of theproposed approach.

Sparse Subspace Clustering via Two-Step Reweighted L1-Minimization: Algorithm and Provable Neighbor Recovery Rates