Structured low-rank matrix learning: algorithms and applications
This work addresses matrix learning problems in machine learning, such as matrix completion and multi-task learning, with incremental improvements in optimization efficiency.
The paper tackles the problem of learning a low-rank matrix with linear subspace constraints by introducing a novel factorization that decouples low-rank and structural constraints, and develops efficient Riemannian optimization algorithms, demonstrating efficacy in experiments on matrix completion and other tasks.
We consider the problem of learning a low-rank matrix, constrained to lie in a linear subspace, and introduce a novel factorization for modeling such matrices. A salient feature of the proposed factorization scheme is it decouples the low-rank and the structural constraints onto separate factors. We formulate the optimization problem on the Riemannian spectrahedron manifold, where the Riemannian framework allows to develop computationally efficient conjugate gradient and trust-region algorithms. Experiments on problems such as standard/robust/non-negative matrix completion, Hankel matrix learning and multi-task learning demonstrate the efficacy of our approach. A shorter version of this work has been published in ICML'18.