Compressive Robust Principal Component Analysis (CRPCA) naturally arises in various applications as a means to recover a low-rank matrix low-rank matrix $\boldsymbol{L}$ and a sparse matrix $\boldsymbol{S}$ from compressive measurements. In this paper, we approach the problem from a Bayesian inference perspective. We establish a probabilistic model for the problem and develop an improved turbo message passing (ITMP) algorithm based on the sum-product rule and the appropriate approximations. Additionally, we establish a state evolution framework to characterize the asymptotic behavior of the ITMP algorithm in the large-system limit. By analyzing the established state evolution, we further propose sufficient conditions for the global convergence of our algorithm. Our numerical results validate the theoretical results, demonstrating that the proposed asymptotic framework accurately characterize the dynamical behavior of the ITMP algorithm, and the phase transition curve specified by the sufficient condition agrees well with numerical simulations.

翻译：压缩鲁棒主成分分析（CRPCA）作为从压缩测量中恢复低秩矩阵$\boldsymbol{L}$与稀疏矩阵$\boldsymbol{S}$的方法，在众多应用场景中自然出现。本文从贝叶斯推断的角度研究该问题。我们建立了该问题的概率模型，并基于和积规则与适当近似，提出了一种改进型Turbo消息传递（ITMP）算法。此外，我们构建了状态演化框架以刻画ITMP算法在大系统极限下的渐近行为。通过分析所建立的状态演化，我们进一步提出了算法全局收敛的充分条件。数值实验结果验证了理论分析，表明所提出的渐近框架能准确描述ITMP算法的动态行为，且充分条件确定的相变曲线与数值模拟结果高度吻合。