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.