论文标题
线性收敛算法，用于旋转不变$ \ ell_1 $ -norm主成分分析
A Linearly Convergent Algorithm for Rotationally Invariant $\ell_1$-Norm Principal Component Analysis
论文作者
论文摘要
为了通过离群值对数据集进行降低，$ \ ell_1 $ -Norm主成分分析（L1-PCA）作为常规PCA的典型稳健替代方案，在过去的几年中一直很有名。在这项工作中，我们考虑了旋转不变的L1-PCA，文献中几乎没有研究。为了解决这个问题，我们提出了一种近端交替的线性化最小化方法，并通过非线性外推，用于解决其两个阻滞重新印象。此外，我们表明，所提出的方法至少线性地收敛到重新计算问题的限制关键点。事实证明，在施加的条件下，这种点被证明是原始问题的关键点。最后，我们对合成数据集和实际数据集进行了数值实验，以支持我们的理论发展，并证明了方法的疗效。
To do dimensionality reduction on the datasets with outliers, the $\ell_1$-norm principal component analysis (L1-PCA) as a typical robust alternative of the conventional PCA has enjoyed great popularity over the past years. In this work, we consider a rotationally invariant L1-PCA, which is hardly studied in the literature. To tackle it, we propose a proximal alternating linearized minimization method with a nonlinear extrapolation for solving its two-block reformulation. Moreover, we show that the proposed method converges at least linearly to a limiting critical point of the reformulated problem. Such a point is proved to be a critical point of the original problem under a condition imposed on the step size. Finally, we conduct numerical experiments on both synthetic and real datasets to support our theoretical developments and demonstrate the efficacy of our approach.
