fLRR: Fast Low-Rank Representation Using Frobenius Norm

Low Rank Representation (LRR) intends to find the representation with lowest-rank of a given data set, which can be formulated as a rank minimization problem. Since the rank operator is non-convex and discontinuous, most of the recent works use the nuclear norm as a convex relaxation. This letter theoretically shows that under some conditions, Frobenius-norm-based optimization problem has an unique solution that is also a solution of the original LRR optimization problem. In other words, it is feasible to apply Frobenius-norm as a surrogate of the nonconvex matrix rank function. This replacement will largely reduce the time-costs for obtaining the lowest-rank solution. Experimental results show that our method (i.e., fast Low Rank Representation, fLRR), performs well in terms of accuracy and computation speed in image clustering and motion segmentation compared with nuclear-norm-based LRR algorithm.