Nonconvex Matrix Completion with Linearly Parameterized Factors
This work addresses matrix completion for applications like collaborative filtering by incorporating structured priors, representing an incremental improvement with theoretical analysis.
The paper tackles the problem of matrix completion with prior information by proposing a nonconvex optimization framework with linearly parameterized factors, establishing theoretical guarantees for local minima and demonstrating effectiveness through numerical simulations.
Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice including collaborative filtering, prior information and special structures are usually employed in order to improve the accuracy of matrix completion. In this paper, we propose a unified nonconvex optimization framework for matrix completion with linearly parameterized factors. In particular, by introducing a condition referred to as Correlated Parametric Factorization, we can conduct a unified geometric analysis for the nonconvex objective by establishing uniform upper bounds for low-rank estimation resulting from any local minimum. Perhaps surprisingly, the condition of Correlated Parametric Factorization holds for important examples including subspace-constrained matrix completion and skew-symmetric matrix completion. The effectiveness of our unified nonconvex optimization method is also empirically illustrated by extensive numerical simulations.