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

TL;DR

This paper introduces novel bilinear factor matrix norm minimization models for robust PCA that are more scalable and accurate, especially in low observation scenarios, and demonstrates their effectiveness in various low-level vision tasks.

Contribution

The paper proposes two new bilinear factor matrix norm models based on Schatten quasi-norms, improving scalability and accuracy for robust PCA in vision applications.

Findings

Our methods outperform traditional Schatten quasi-norm minimization.

The proposed models are more scalable and suitable for large-scale problems.

Applications show superior performance in tasks like text removal and image inpainting.

Abstract

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.