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We consider the problem of learning a low-rank matrix, constrained to lie ina linear subspace, and introduce a novel factorization for modeling suchmatrices. A salient feature of the proposed factorization scheme is itdecouples the low-rank and the structural constraints onto separate factors. Weformulate the optimization problem on the Riemannian spectrahedron manifold,where the Riemannian framework allows to develop computationally efficientconjugate gradient and trust-region algorithms. Experiments on problems such asstandard/robust/non-negative matrix completion, Hankel matrix learning andmulti-task learning demonstrate the efficacy of our approach. A shorter versionof this work has been published in ICML'18.

Structured low-rank matrix learning: algorithms and applications