Symmetric Matrix Completion with ReLU Sampling
This addresses matrix completion challenges in scenarios with entry-dependent sampling, such as recommendation systems or sensor networks, but is incremental as it builds on existing nonconvex optimization frameworks.
The paper tackles the problem of symmetric positive semi-definite low-rank matrix completion with ReLU sampling, where only positive entries are observed, and proves that under mild assumptions, the nonconvex objective is geodesically strongly convex near the planted matrix, enabling global convergence with a tailored initialization.
We study the problem of symmetric positive semi-definite low-rank matrix completion (MC) with deterministic entry-dependent sampling. In particular, we consider rectified linear unit (ReLU) sampling, where only positive entries are observed, as well as a generalization to threshold-based sampling. We first empirically demonstrate that the landscape of this MC problem is not globally benign: Gradient descent (GD) with random initialization will generally converge to stationary points that are not globally optimal. Nevertheless, we prove that when the matrix factor with a small rank satisfies mild assumptions, the nonconvex objective function is geodesically strongly convex on the quotient manifold in a neighborhood of a planted low-rank matrix. Moreover, we show that our assumptions are satisfied by a matrix factor with i.i.d. Gaussian entries. Finally, we develop a tailor-designed initialization for GD to solve our studied formulation, which empirically always achieves convergence to the global minima. We also conduct extensive experiments and compare MC methods, investigating convergence and completion performance with respect to initialization, noise level, dimension, and rank.