# Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning

arXiv: https://arxiv.org/abs/2203.02189 · HTML: https://mlsanity.com/paper/2203.02189/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 51 references that are in the map.
- First posted on arXiv 4 Mar 2022.

## Builds on (most cited first, 7 of 7)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Unsupervised Visual Representation Learning by Context Prediction](https://mlsanity.com/paper/1505.05192/) (2015)
- [Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization](https://mlsanity.com/paper/1708.04181/) (2017)
- [Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications](https://mlsanity.com/paper/1810.05186/) (2018)
- [Factor Group-Sparse Regularization for Efficient Low-Rank Matrix Recovery](https://mlsanity.com/paper/1911.05774/) (2019)
- [A Survey on Matrix Completion: Perspective of Signal Processing](https://mlsanity.com/paper/1901.10885/) (2019)
- [Learning Latent Features with Pairwise Penalties in Low-Rank Matrix Completion](https://mlsanity.com/paper/1802.05821/) (2018)

## Built on by (most cited first, 4 of 4)

- [Positive-incentive Noise](https://mlsanity.com/paper/2212.09541/) (2022)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [Matrix Completion via Nonsmooth Regularization of Fully Connected Neural Networks](https://mlsanity.com/paper/2403.10232/) (2024)
- [Generalized Low-Rank Matrix Completion Model with Overlapping Group Error Representation](https://mlsanity.com/paper/2407.08517/) (2024)

## Abstract (arXiv)

The existing matrix completion methods focus on optimizing the relaxation of rank function such as nuclear norm, Schatten-p norm, etc. They usually need many iterations to converge. Moreover, only the low-rank property of matrices is utilized in most existing models and several methods that incorporate other knowledge are quite time-consuming in practice. To address these issues, we propose a novel non-convex surrogate that can be optimized by closed-form solutions, such that it empirically converges within dozens of iterations. Besides, the optimization is parameter-free and the convergence is proved. Compared with the relaxation of rank, the surrogate is motivated by optimizing an upper-bound of rank. We theoretically validate that it is equivalent to the existing matrix completion models. Besides the low-rank assumption, we intend to exploit the column-wise correlation for matrix completion, and thus an adaptive correlation learning, which is scaling-invariant, is developed. More importantly, after incorporating the correlation learning, the model can be still solved by closed-form solutions such that it still converges fast. Experiments show the effectiveness of the non-convex surrogate and adaptive correlation learning.