Nonlinear matrix recovery using optimization on the Grassmann manifold
This work addresses matrix recovery for nonlinear structures, which is an incremental advancement in optimization methods for specific data types like unions of subspaces or clustering.
The paper tackles the problem of recovering partially observed high-rank matrices with nonlinear structures, such as unions of subspaces or clusters, by formulating it as a rank minimization problem and solving it with optimization on the Grassmann manifold. The result includes theoretical guarantees for convergence and competitive numerical performance, with high accuracy achieved using Riemannian second-order methods.
We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.