# Robust Low-rank Matrix Completion via an Alternating Manifold Proximal Gradient Continuation Method

arXiv: https://arxiv.org/abs/2008.07740 · HTML: https://mlsanity.com/paper/2008.07740/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Aug 2020.

## Built on by (most cited first, 5 of 5)

- [Smooth Robust Tensor Completion for Background/Foreground Separation with Missing Pixels: Novel Algorithm with Convergence Guarantee](https://mlsanity.com/paper/2203.16328/) (2022)
- [Bayesian Low-rank Matrix Completion with Dual-graph Embedding: Prior Analysis and Tuning-free Inference](https://mlsanity.com/paper/2203.10044/) (2022)
- [Streaming data preprocessing via online tensor recovery for large environmental sensor networks](https://mlsanity.com/paper/2109.00596/) (2021)
- [RGNMR: A Gauss-Newton method for robust matrix completion with theoretical guarantees](https://mlsanity.com/paper/2505.12919/) (2025)
- [High Rank Matrix Completion via Grassmannian Proxy Fusion](https://mlsanity.com/paper/2602.02565/) (2026)

## Abstract (arXiv)

Robust low-rank matrix completion (RMC), or robust principal component analysis with partially observed data, has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims to decompose a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) and the l1 norm of the sparse matrix (to promote sparsity). In this paper, motivated by some recent works on low-rank matrix completion and Riemannian optimization, we formulate this problem as a nonsmooth Riemannian optimization problem over Grassmann manifold. This new formulation is scalable because the low-rank matrix is factorized to the multiplication of two much smaller matrices. We then propose an alternating manifold proximal gradient continuation (AManPGC) method to solve the proposed new formulation. The convergence rate of the proposed algorithm is rigorously analyzed. Numerical results on both synthetic data and real data on background extraction from surveillance videos are reported to demonstrate the advantages of the proposed new formulation and algorithm over several popular existing approaches.