Mathematical Programming Computation, Volume 12, Issue 4, December 2020
A multi-stage convex relaxation approach to noisy structured low-rank matrix recovery
Shujun Bi, Shaohua Pan, Defeng Sun
This paper concerns with a noisy structured low-rank matrix recovery problem which can be
modeled as a structured rank minimization problem. We reformulate this problem as a mathematical
program with a generalized complementarity constraint (MPGCC), and show that its penalty version,
yielded by moving the generalized complementarity constraint to the objective, has the same global
optimal solution set as the MPGCC does whenever the penalty parameter is over a certain threshold.
Then, by solving the exact penalty problem in an alternating way, we obtain a multi-stage convex
relaxation approach. We provide theoretical guarantees for our approach under a mild restricted
eigenvalue condition, by quantifying the reduction of the error and approximate rank bounds of
the first stage convex relaxation in the subsequent stages and establishing the geometric
convergence of the error sequence in a statistical sense. Numerical experiments are conducted
for some structured low-rank matrix recovery examples to confirm our theoretical findings.
Our code can be achieved from https://doi.org/10.5281/zenodo.3600639.
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