# Tensor factorization based method for low rank matrix completion and its application on tensor completion

arXiv: https://arxiv.org/abs/2201.09197 · HTML: https://mlsanity.com/paper/2201.09197/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2022.

## Built on by (most cited first, 2 of 2)

- [Spectral-Spatial Extraction through Layered Tensor Decomposition for Hyperspectral Anomaly Detection](https://mlsanity.com/paper/2503.05183/) (2025)
- [Color Image Recovery Using Generalized Matrix Completion over Higher-Order Finite Dimensional Algebra](https://mlsanity.com/paper/2308.02621/) (2023)

## Abstract (arXiv)

Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data by using their low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency. To this end, we first establish the relationship between matrix rank and tensor tubal rank, and then reformulate matrix completion problem as a tensor completion problem. For the reformulated tensor completion problem, we adopt a two-stage strategy based on tensor factorization algorithm. In this way, a matrix completion problem of big size can be solved via some matrix computations of smaller sizes. For a third order tensor completion problem, to fully exploit the low rank structures, we introduce the double tubal rank which combines the tubal rank and the rank of the mode-3 unfolding matrix. For the mode-3 unfolding matrix rank, we follow the idea of matrix completion. Based on this, we establish a novel model and modify the tensor factorization based algorithm for third order tensor completion. Extensive numerical experiments demonstrate that the proposed methods outperform state-of-the-art methods in terms of both accuracy and running time.