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A general framework for solving the subspace clustering problem using the CURdecomposition is presented. The CUR decomposition provides a natural way toconstruct similarity matrices for data that come from a union of unknownsubspaces $\mathscr{U}=\underset{i=1}{\overset{M}\bigcup}S_i$. The similaritymatrices thus constructed give the exact clustering in the noise-free case.Additionally, this decomposition gives rise to many distinct similaritymatrices from a given set of data, which allow enough flexibility to performaccurate clustering of noisy data. We also show that two known methods forsubspace clustering can be derived from the CUR decomposition. An algorithmbased on the theoretical construction of similarity matrices is presented, andexperiments on synthetic and real data are presented to test the method.  Additionally, an adaptation of our CUR based similarity matrices is utilizedto provide a heuristic algorithm for subspace clustering; this algorithm yieldsthe best overall performance to date for clustering the Hopkins155 motionsegmentation dataset.

CUR Decompositions, Similarity Matrices, and Subspace Clustering