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We propose a convex optimization formulation with the Ky Fan $2$-$k$-norm and$\ell_1$-norm to find $k$ largest approximately rank-one submatrix blocks of agiven nonnegative matrix that has low-rank block diagonal structure with noise.We analyze low-rank and sparsity structures of the optimal solutions usingproperties of these two matrix norms. We show that, under certain hypotheses,with high probability, the approach can recover rank-one submatrix blocks evenwhen they are corrupted with random noise and inserted into a much largermatrix with other random noise blocks.

Finding the largest low-rank clusters with Ky Fan $2$-$k$-norm and $\ell_1$-norm