# Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications

arXiv: https://arxiv.org/abs/1810.05186 · HTML: https://mlsanity.com/paper/1810.05186/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2018.

## Built on by (most cited first, 16 of 16)

- [Smooth Robust Tensor Completion for Background/Foreground Separation with Missing Pixels: Novel Algorithm with Convergence Guarantee](https://mlsanity.com/paper/2203.16328/) (2022)
- [Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning](https://mlsanity.com/paper/2203.02189/) (2022)
- [A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds](https://mlsanity.com/paper/2104.04199/) (2021)
- [Quaternion Optimized Model with Sparse Regularization for Color Image Recovery](https://mlsanity.com/paper/2204.08629/) (2022)
- [A Novel Tensor Factorization-Based Method with Robustness to Inaccurate Rank Estimation](https://mlsanity.com/paper/2305.11458/) (2023)
- [FADE: A Dataset for Detecting Falling Objects around Buildings in Video](https://mlsanity.com/paper/2408.05750/) (2024)
- [Efficient Differentiable Approximation of Generalized Low-rank Regularization](https://mlsanity.com/paper/2505.15407/) (2025)
- [Inference and Optimization for Engineering and Physical Systems](https://mlsanity.com/paper/2208.13880/) (2022)
- [Region-wise matching for image inpainting based on adaptive weighted low-rank decomposition](https://mlsanity.com/paper/2303.12421/) (2023)
- [L2,1-Norm Regularized Quaternion Matrix Completion Using Sparse Representation and Quaternion QR Decomposition](https://mlsanity.com/paper/2309.03764/) (2023)
- [Robust Low-Rank Matrix Completion via a New Sparsity-Inducing Regularizer](https://mlsanity.com/paper/2310.04762/) (2023)
- [Robust matrix completion via Novel M-estimator Functions](https://mlsanity.com/paper/2310.04953/) (2023)
- [A framework to generate sparsity-inducing regularizers for enhanced low-rank matrix completion](https://mlsanity.com/paper/2310.04954/) (2023)
- [Depth-Aided Color Image Inpainting in Quaternion Domain](https://mlsanity.com/paper/2503.16818/) (2025)
- [Matrix Completion Via Reweighted Logarithmic Norm Minimization](https://mlsanity.com/paper/2512.21050/) (2025)
- [Robust Principal Component Completion](https://mlsanity.com/paper/2603.25132/) (2026)

## Abstract (arXiv)

The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to lp-norm minimization with two specific values of p, i.e., p=1/2 and p=2/3, we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis. We first define the double nuclear norm and Frobenius/nuclear hybrid norm penalties, and then prove that they are in essence the Schatten-1/2 and 2/3 quasi-norms, respectively, which lead to much more tractable and scalable Lipschitz optimization problems. Our experimental analysis shows that both our methods yield more accurate solutions than original Schatten quasi-norm minimization, even when the number of observations is very limited. Finally, we apply our penalties to various low-level vision problems, e.g., text removal, moving object detection, image alignment and inpainting, and show that our methods usually outperform the state-of-the-art methods.