The paper proposes a novel method for subspace clustering based on sparse representation (SR). The method leverages the fact that each point in a union of subspaces can be represented as a sparse combination of other points. The key contribution is that under mild assumptions, this sparse representation can be obtained efficiently using $\ell_1$ optimization. The segmentation of the data is then achieved by applying spectral clustering to a similarity matrix built from the sparse representation. The method handles noise, outliers, and missing data, and is evaluated on the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that the proposed method significantly outperforms state-of-the-art methods in terms of accuracy and robustness.The paper proposes a novel method for subspace clustering based on sparse representation (SR). The method leverages the fact that each point in a union of subspaces can be represented as a sparse combination of other points. The key contribution is that under mild assumptions, this sparse representation can be obtained efficiently using $\ell_1$ optimization. The segmentation of the data is then achieved by applying spectral clustering to a similarity matrix built from the sparse representation. The method handles noise, outliers, and missing data, and is evaluated on the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that the proposed method significantly outperforms state-of-the-art methods in terms of accuracy and robustness.