# Efficient enhancement of low-rank tensor completion via thin QR decomposition

## TL;DR

This paper introduces a more efficient method for completing missing data in tensors using thin QR decomposition instead of traditional methods.

## Contribution

The novel contribution is a CTNM method using thin QR decomposition, reducing computational cost and improving accuracy.

## Key findings

- The CTNM-QR method reduces computational complexity by replacing SVD with thin QR decomposition.
- Experiments show CTNM-QR outperforms existing methods in accuracy and efficiency on synthetic and real data.
- The proposed method achieves better tensor completion results in color images and brain MRI data.

## Abstract

Low-rank tensor completion (LRTC), which aims to complete missing entries from tensors with partially observed terms by utilizing the low-rank structure of tensors, has been widely used in various real-world issues. The core tensor nuclear norm minimization (CTNM) method based on Tucker decomposition is one of common LRTC methods. However, the CTNM methods based on Tucker decomposition often have a large computing cost due to the fact that the general factor matrix solving technique involves multiple singular value decompositions (SVDs) in each loop. To address this problem, this article enhances the method and proposes an effective CTNM method based on thin QR decomposition (CTNM-QR) with lower computing complexity. The proposed method extends the CTNM by introducing tensor versions of the auxiliary variables instead of matrices, while using the thin QR decomposition to solve the factor matrix rather than the SVD, which can save the computational complexity and improve the tensor completion accuracy. In addition, the CTNM-QR method's convergence and complexity are analyzed further. Numerous experiments in synthetic data, real color images, and brain MRI data at different missing rates demonstrate that the proposed method not only outperforms in terms of completion accuracy and visualization, but also conducts more efficiently than most state-of-the-art LRTC methods.