Title: Noisy $\ell^{0}$-Sparse Subspace Clustering on Dimensionality Reduced Data. Abstract: High-dimensional data often lie in or close to low-dimensional subspaces. Sparse subspace clustering methods with sparsity induced by L0-norm, such as L0-Sparse Subspace Clustering (L0-SSC), are demonstrated to be more effective than its L1 counterpart such as Sparse Subspace Clustering (SSC). However, these L0-norm based subspace clustering methods are restricted to clean data that lie exactly in subspaces. Real data often suffer from noise and they may lie close to subspaces. We propose noisy L0-SSC to handle noisy data so as to improve the robustness. We show that the optimal solution to the optimization problem of noisy L0-SSC achieves subspace detection property (SDP), a key element with which data from different subspaces are separated, under deterministic and randomized models. Our results provide theoretical guarantee on the correctness of noisy L0-SSC in terms of SDP on noisy data. We further propose Noisy-DR-L0-SSC which provably recovers the subspaces on dimensionality reduced data. Noisy-DR-L0-SSC first projects the data onto a lower dimensional space by linear transformation, then performs noisy L0-SSC on the dimensionality reduced data so as to improve the efficiency. The experimental results demonstrate the effectiveness of noisy L0-SSC and Noisy-DR-L0-SSC.