Sketching low-rank matrices with a shared column space by convex programming
share
In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such matrices from their noisy linear measurements. We study a convex estimator regularized by a pair of matrix norms. The measurement model corresponds to block-wise sensing and the reconstruction is possible only when the total energy is well distributed over blocks. The first norm, which is the maximum-block-Frobenius norm, favors such a solution. This condition is analogous to the notion of incoherence in matrix completion or column-wise sensing. The second norm, which is a tensor norm on a pair of suitable Banach spaces, induces low-rankness in the solution together with the first norm. We demonstrate that the convex estimator provides a provably near optimal error bound that matches a minimax lower bound up to a logarithmic factor. The convex estimator is cast as a semidefinite program and an efficient ADMM-based algorithm is derived. The empirical behavior of the convex estimator is illustrated using Monte Carlo simulations and recovery performance is compared to existing methods in the literature.
READ FULL TEXT