# A regularized deep matrix factorized model of matrix completion for image restoration

arXiv: https://arxiv.org/abs/2007.14581 · HTML: https://mlsanity.com/paper/2007.14581/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2020.

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

- [Simple, Fast, and Flexible Framework for Matrix Completion with Infinite Width Neural Networks](https://mlsanity.com/paper/2108.00131/) (2021)
- [Regularize implicit neural representation by itself](https://mlsanity.com/paper/2303.15484/) (2023)
- [Adaptive and Implicit Regularization for Matrix Completion](https://mlsanity.com/paper/2208.05640/) (2022)
- [AIR-Net: Adaptive and Implicit Regularization Neural Network for Matrix Completion](https://mlsanity.com/paper/2110.07557/) (2021)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)

## Abstract (arXiv)

It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low-rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorization component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. However, low rank is not adequate to reflect the intrinsic characteristics of a natural image. Thus, algorithms with only the constraint of low rank are insufficient to perform image restoration well. In this work, we propose a Regularized Deep Matrix Factorized (RDMF) model for image restoration, which utilizes the implicit bias of the low rank of deep neural networks and the explicit bias of total variation. We demonstrate the effectiveness of our RDMF model with extensive experiments, in which our method surpasses the state of art models in common examples, especially for the restoration from very few observations. Our work sheds light on a more general framework for solving other inverse problems by combining the implicit bias of deep learning with explicit regularization.