# Towards Faster Rates and Oracle Property for Low-Rank Matrix Estimation

arXiv: https://arxiv.org/abs/1505.04780 · HTML: https://mlsanity.com/paper/1505.04780/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2015.

## Built on by (most cited first, 3 of 3)

- [Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization](https://mlsanity.com/paper/2205.03059/) (2022)
- [Low-Rank Tensor Learning by Generalized Nonconvex Regularization](https://mlsanity.com/paper/2410.18402/) (2024)
- [Robust Tensor Regression with Nonconvexity: Algorithmic and Statistical Theory](https://mlsanity.com/paper/2605.07448/) (2026)

## Abstract (arXiv)

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.