CUR Decompositions, Similarity Matrices, and Subspace Clustering
This work addresses subspace clustering, a key problem in machine learning for data analysis, with incremental improvements over existing methods.
The paper tackles subspace clustering by introducing a framework based on CUR decomposition to construct similarity matrices, which achieve exact clustering in noise-free cases and flexibility for noisy data. Experiments show the method yields the best performance to date on the Hopkins155 motion segmentation dataset.
A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces $\mathscr{U}=\underset{i=1}{\overset{M}\bigcup}S_i$. The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method. Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.