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

TL;DR

This paper introduces a convex optimization method combining Ky Fan 2-$k$-norm and $\,ell_1$-norm to accurately identify the largest low-rank, approximately rank-one submatrix blocks within noisy, block-structured matrices.

Contribution

It presents a novel convex formulation leveraging specific matrix norms to recover low-rank blocks in noisy matrices, with theoretical guarantees under certain conditions.

Findings

Method successfully recovers rank-one blocks with high probability.

Approach is robust to random noise within the matrix.

Theoretical analysis supports the effectiveness of the convex formulation.

Abstract

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 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.