Subspace clustering with first-order log-determinant rank approximation

It is natural to assume a data matrix to be of low-rank in many machine learning and computer vision problems, e.g., matrix completion and subspace clustering. To ensure low-rank, the nuclear norm minimization is widely used as a rank approximation and minimization technique. The nuclear norm treats all the singular values through adding them together with equal weights. With this liner addition, large singulars will con-tribute too much and make the approximation very inaccurate. To reduce this undesirable effect, we make use of a first-order log-determinant approximation which assigns small weights to large singular values and is close to vanishing for small ones. A low-rank representation is obtained using this function and the affinity is then constructed based on angular in-formation. Experimentally we achieve promising results on face clustering and motion segmentation using this affinity.