  Subspace clustering (SC) is a popular method for dimensionality reduction of
high-dimensional data, where it generalizes Principal Component Analysis (PCA).
Recently, several methods have been proposed to enhance the robustness of PCA
and SC, while most of them are computationally very expensive, in particular,
for high dimensional large-scale data. In this paper, we develop much faster
iterative algorithms for SC, incorporating robustness using a {\em non-squared}
$\ell_2$-norm objective. The known implementations for optimizing the objective
would be costly due to the alternative optimization of two separate objectives:
optimal cluster-membership assignment and robust subspace selection, while the
substitution of one process to a faster surrogate can cause failure in
convergence. To address the issue, we use a simplified procedure requiring
efficient matrix-vector multiplications for subspace update instead of solving
an expensive eigenvector problem at each iteration, in addition to release
nested robust PCA loops. We prove that the proposed algorithm monotonically
converges to a local minimum with approximation guarantees, e.g., it achieves
2-approximation for the robust PCA objective. In our experiments, the proposed
algorithm is shown to converge at an order of magnitude faster than known
algorithms optimizing the same objective, and have outperforms prior subspace
clustering methods in accuracy and running time for MNIST dataset.
