Robust principal component analysis (RPCA) has recently seen ubiquitous activity for dimensionality reduction in image processing, visualization and pattern recognition. Conventional RPCA methods model the low-rank component as regularizing each singular value equally. However, in numerous modern applications, each singular value has different physical meaning and should be treated differently. This is one of the main reasons why RPCA techniques cannot work well in dealing with many realistic problems. To solve this problem, a novel hierarchical Bayesian RPCA model with adaptive singular value penalty is proposed. This model enforces the low-rank constraint by introducing an adaptive penalty function on the singular values of the low-rank component. In particular, we impose a hierarchical Exponent-Gamma prior on the singular values of the low-rank component and the Beta-Bernoulli prior on sparsity indicators. The variational Bayesian framework and the Markov chain Monte Carlo-based Bayesian inference are considered for inferring the posteriors of all latent variables involved in low-rank and sparse components. Numerical experiments demonstrate the competitive performance of the proposed model on synthetic and real data.

中文翻译：

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具有自适应奇异值惩罚的贝叶斯稳健主成分分析

稳健主成分分析 (RPCA) 最近在图像处理、可视化和模式识别中发现了无处不在的降维活动。传统的 RPCA 方法将低秩分量建模为对每个奇异值均等地进行正则化。然而，在众多现代应用中，每个奇异值都有不同的物理意义，应该区别对待。这是 RPCA 技术无法很好地处理许多现实问题的主要原因之一。为了解决这个问题，提出了一种具有自适应奇异值惩罚的新型分层贝叶斯RPCA模型。该模型通过对低秩分量的奇异值引入自适应惩罚函数来强制执行低秩约束。特别是，我们对低秩分量的奇异值和稀疏指标的 Beta-Bernoulli 先验强加了分层指数伽玛先验。变分贝叶斯框架和基于马尔可夫链蒙特卡罗的贝叶斯推理被考虑用于推断低秩和稀疏分量中涉及的所有潜在变量的后验。数值实验证明了所提出的模型在合成和真实数据上的竞争性能。