  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.

Machine Learning;Computer Vision and Pattern Recognition;Optimization and Control;Machine Learning