Towards Faster Rates and Oracle Property for Low-Rank Matrix Estimation
This work addresses matrix completion problems for data analysis, offering theoretical advancements but is incremental as it builds on existing nonconvex penalty methods.
The paper tackles low-rank matrix estimation by proposing a nonconvex penalty framework, proving it achieves a faster statistical rate and exact rank recovery under certain conditions, with numerical validation on synthetic and real datasets.
We present a unified framework for low-rank matrix estimation with nonconvex penalties. We first prove that the proposed estimator attains a faster statistical rate than the traditional low-rank matrix estimator with nuclear norm penalty. Moreover, we rigorously show that under a certain condition on the magnitude of the nonzero singular values, the proposed estimator enjoys oracle property (i.e., exactly recovers the true rank of the matrix), besides attaining a faster rate. As far as we know, this is the first work that establishes the theory of low-rank matrix estimation with nonconvex penalties, confirming the advantages of nonconvex penalties for matrix completion. Numerical experiments on both synthetic and real world datasets corroborate our theory.