Matrix Completion with Model-free Weighting

Abstract

In this paper, we propose a novel method for matrix completion under general nonuniform missing structures. By controlling an upper bound of a novel balancing error, we construct weights that can actively adjust for the non-uniformity in the empirical risk without explicitly modeling the observation probabilities, and can be computed efficiently via convex optimization. The recovered matrix based on the proposed weighted empirical risk enjoys appealing theoretical guarantees. In particular, the proposed method achieves stronger guarantee than existing work in terms of the scaling with respect to the observation probabilities, under asymptotically heterogeneous missing settings (where entry-wise observation probabilities can be of different orders). These settings can be regarded as a better theoretical model of missing patterns with highly varying probabilities. We also provide a new minimax lower bound under a class of heterogeneous settings. Numerical experiments are also provided to demonstrate the effectiveness of the proposed method.

Generalization Bounds for Inductive Matrix Completion in Low-Noise Settings

“…For instance, some work on max norm regularisation has this property (Cai and Zhou 2016). Some results of order r O `σ a nr N ˘were also obtained for matrix completion with a special algorithm that requires explicit rank restriction (Keshavan, Montanari, and Oh 2009; Wang et al 2021). Approximate Recovery Results: There is a wide body of works proving uniform-convergence type generalization bounds for various matrix completion settings.…”