Finding the largest low-rank clusters with Ky Fan 2-k-norm and l1-norm

We propose a convex optimization formulation with the Ky Fan 2-k-norm and l1-norm to find k largest approximately rank-one submatrix blocks of a given nonnegative matrix that has low-rank block diagonal structure with noise. We analyze low-rank and sparsity structures of the optimal solutions using properties of these two matrix norms. We show that, under certain hypotheses, with high probability, the approach can recover rank-one submatrix blocks even when they are corrupted with random noise and inserted into a much larger matrix with other random noise blocks

A proximal point algorithm for sequential feature extraction applications

We propose a proximal point algorithm to solve the LAROS problem, that is, the problem of finding a "large approximately rank-one submatrix." This LAROS problem is used to sequentially extract features in data. We also develop new stopping criteria for the proximal point algorithm, which is based on the duality conditions of epsilon-optimal solutions of the LAROS problem, with a theoretical guarantee. We test our algorithm with two image databases and show that we can use the LAROS problem to extract appropriate common features from these images