# A multi-stage convex relaxation approach to noisy structured low-rank matrix recovery

## TL;DR

This paper introduces a multi-stage convex relaxation method for noisy structured low-rank matrix recovery, providing theoretical guarantees and demonstrating its effectiveness through numerical experiments.

## Contribution

It proposes a novel multi-stage convex relaxation approach based on an exact penalty formulation, with proven convergence and error reduction guarantees.

## Key findings

- The method achieves geometric convergence of the error sequence.
- The approach outperforms standard nuclear norm relaxation in experiments.
- Theoretical analysis confirms error reduction and rank approximation bounds.

## Abstract

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 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 (which is exactly the nuclear norm 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.