Robust Principal Component Analysis via Truncated $L_{1-2}$ Minimization

Robust principal component analysis (RPCA) has gained popularity for handling high-dimensional data. The nuclear norm minimization (NNM) in RPCA is a classical method and has been widely investigated, which can recover low-rank and sparse matrices with high probability under certain conditions. However, NNM shrinks all singular values by the same threshold and over-penalizes larger singular values, resulting in this model being biased. Therefore, we propose a new method based on the truncated $l_{1-2}$ norm to solve this problem in this paper, which is unbiased and flexible to capture the low-rank structure of the data matrix more accurately while separating the sparse noise. We also develop a robust and efficient algorithm to solve the proposed nonconvex optimization model, with the computational complexity and convergence discussed. Then the proposed scheme is applied to synthetic data as well as real-world data, including video background subtraction, facial shadow removal, and anomaly detection, for testing. These experimental results demonstrate that our proposed method is effect and superior in accuracy and robustness compared to other state-of-the-art methods.