  Subspace clustering and feature extraction are two of the most commonly used
unsupervised learning techniques in computer vision and pattern recognition.
State-of-the-art techniques for subspace clustering make use of recent advances
in sparsity and rank minimization. However, existing techniques are
computationally expensive and may result in degenerate solutions that degrade
clustering performance in the case of insufficient data sampling. To partially
solve these problems, and inspired by existing work on matrix factorization,
this paper proposes fixed-rank representation (FRR) as a unified framework for
unsupervised visual learning. FRR is able to reveal the structure of multiple
subspaces in closed-form when the data is noiseless. Furthermore, we prove that
under some suitable conditions, even with insufficient observations, FRR can
still reveal the true subspace memberships. To achieve robustness to outliers
and noise, a sparse regularizer is introduced into the FRR framework. Beyond
subspace clustering, FRR can be used for unsupervised 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 benefits of FRR for unsupervised
visual learning.
