Fixed-Rank Representation for Unsupervised Visual Learning

Subspace clustering and feature extraction are two of the most extensive unsupervised visual learning tasks in computer vision and pattern recognition. In this paper, we pose these two problems in a unified framework, named fixed-rank representation (FRR). For subspace clustering, our first contribution is to show that, when the data is clean, we can efficiently solve FRR in closed-form and the global optimal solution to FRR can exactly recover the multiple subspace structure. Furthermore, we prove that under some suitable conditions, even with insufficient observations, the memberships of data points still can be exactly recovered by FRR. In the case that the data is corrupted by noises and outliers, a sparse regularization is introduced to achieve robustness for FRR. For feature extraction, we provide some new insights to understand existing methods, which lead to a new approach for robust feature extraction. As a non-trivial byproduct, a fast numerical solver is developed for FRR. Experimental results on both synthetic data and real applications validate our theoretical analysis and demonstrate the power of FRR for unsupervised visual learning.