This paper proposes a novel method called Low-Rank Representation (LRR) for robust subspace segmentation. Unlike sparse representation (SR), which finds the sparsest representation of each data vector individually, LRR aims to find the lowest-rank representation of a collection of vectors jointly. LRR better captures the global structure of data, making it more effective for robust subspace segmentation from corrupted data. Theoretical and experimental results show that LRR is a promising tool for subspace segmentation. LRR represents data vectors as linear combinations of other vectors in a dictionary. Given a set of data vectors drawn from a union of multiple subspaces, LRR finds the lowest-rank representation of all data jointly. This representation can be used to define the affinities of an undirected graph, and the final segmentation results can be obtained by spectral clustering. Compared to SR, LRR is better at capturing global structures of data, making it more suitable for subspace segmentation. When subspaces are independent, LRR can reveal the membership of the samples: within-cluster affinities are dense, and between-cluster affinities are all zeros. This solution solves a nuclear norm minimization problem and provides an efficient algorithm for solving it. For corrupted data, the lowest-rank criterion can correct corruptions, making LRR robust to noise and outliers. The paper also discusses the robustness of LRR to noise and outliers. It introduces a modified version of LRR that incorporates an $ \ell_{2,1} $-norm to handle corruptions. This approach assumes that corruptions are "sample-specific," i.e., some data vectors are corrupted and others are clean. The results show that LRR can effectively handle corruptions and is more robust than other methods like SR, GPCA, LSA, RANSAC, and SSC. Experiments on toy data, slightly corrupted data, and heavily corrupted data demonstrate the effectiveness of LRR. It outperforms other methods in terms of segmentation accuracy, especially in the presence of corruptions. The paper also discusses future work, including learning a compact dictionary for LRR, extending its application to supervised classification, and establishing theoretical conditions for the success of data recovery from multiple subspaces.This paper proposes a novel method called Low-Rank Representation (LRR) for robust subspace segmentation. Unlike sparse representation (SR), which finds the sparsest representation of each data vector individually, LRR aims to find the lowest-rank representation of a collection of vectors jointly. LRR better captures the global structure of data, making it more effective for robust subspace segmentation from corrupted data. Theoretical and experimental results show that LRR is a promising tool for subspace segmentation. LRR represents data vectors as linear combinations of other vectors in a dictionary. Given a set of data vectors drawn from a union of multiple subspaces, LRR finds the lowest-rank representation of all data jointly. This representation can be used to define the affinities of an undirected graph, and the final segmentation results can be obtained by spectral clustering. Compared to SR, LRR is better at capturing global structures of data, making it more suitable for subspace segmentation. When subspaces are independent, LRR can reveal the membership of the samples: within-cluster affinities are dense, and between-cluster affinities are all zeros. This solution solves a nuclear norm minimization problem and provides an efficient algorithm for solving it. For corrupted data, the lowest-rank criterion can correct corruptions, making LRR robust to noise and outliers. The paper also discusses the robustness of LRR to noise and outliers. It introduces a modified version of LRR that incorporates an $ \ell_{2,1} $-norm to handle corruptions. This approach assumes that corruptions are "sample-specific," i.e., some data vectors are corrupted and others are clean. The results show that LRR can effectively handle corruptions and is more robust than other methods like SR, GPCA, LSA, RANSAC, and SSC. Experiments on toy data, slightly corrupted data, and heavily corrupted data demonstrate the effectiveness of LRR. It outperforms other methods in terms of segmentation accuracy, especially in the presence of corruptions. The paper also discusses future work, including learning a compact dictionary for LRR, extending its application to supervised classification, and establishing theoretical conditions for the success of data recovery from multiple subspaces.