卷积鲁棒主成分分析

鲁棒主成分分析（RPCA）是一种经典的高维数据分析方法，可从带噪声的观测样本中恢复出原始数据。但是，RPCA能工作的前提是目标数据拥有低秩矩阵结构，不能有效处理实际应用中广泛存在的非低秩数据。研究发现，虽然图像、视频等数据矩阵本身可能不是低秩的，但它们的卷积矩阵通常是低秩的。根据这一原理，提出一种称为卷积鲁棒主成分分析（CRPCA）的新方法，利用卷积矩阵的低秩性对原始数据的结构进行约束，从而实现精确的数据恢复。CPRCA模型的计算过程是一个凸优化问题，通过乘子交替方向法（ADMM）来进行求解。通过对合成数据向量以及真实数据图片、视频序列进行实验，验证了该方法相较于其他算法如RPCA、广义鲁棒主成分分析（GRPCA）以及核鲁棒主成分分析（KRPCA）在处理数据非低秩问题上优越性。

Convolution robust principal component analysis

Robust Principal Component Analysis (RPCA) is a classical high-dimensional data analysis method, which can recover original data from noisy observation samples. However, the premise that RPCA can work is that the target data has a low rank matrix structure, so that RPCA cannot effectively deal with non-low rank data in practical applications. It is found that the convolution matrices of image and video are usually low rank, although the data matrices of them may not be low rank. According to this principle, a new method called Convolution Robust Principal Component Analysis (CRPCA) was proposed to use the low rank property of convolution matrix to constrain the original data structure, so as to achieve accurate data recovery. The calculation process of CRPCA model is a convex optimization problem, which was solved by Alternating Direction Method of Multipliers (ADMM). Experimental results on synthetic data vectors, real data images and video sequences show that the proposed method is superior to other algorithms such as RPCA, Generalized Robust Principal Component Analysis (GRPCA) and Kernel Robust Principal Component Analysis (KRPCA) in dealing with non-low rank problems.