Global convergence of the alternating-minimization meta-algorithm

Establish whether the alternating-minimization meta-algorithm for jointly estimating the system matrix and binary inlier indicators in the relaxed least-trimmed-squares formulations converges from an arbitrary random initialization, rather than only from an initialization sufficiently close to the ground-truth system matrix.

Background

The paper introduces several continuous relaxations of a combinatorial least-trimmed-squares formulation for robustly learning a strictly stable linear dynamical system from a single trajectory with adversarially corrupted states. The proposed meta-algorithm alternates between estimating the binary inlier indicators through one of the relaxations and estimating the system matrix through constrained weighted least squares.

Because the relaxations are non-convex, the authors note that convergence may depend on the initial estimate of the system matrix. They indicate that an initialization in a small neighborhood of the ground truth might lead to convergence, but leave unresolved whether the same behavior holds for arbitrary random initialization.