基于距离度量的鲁棒主成分分析低维表征算法研究

Low-dimensional representation algorithms for robust principal component analysis based on distance metrics

摘要:

主成分分析（principal component analysis, PCA）作为经典的数据分析和降维方法，在图像压缩、特征提取等领域得到广泛应用。然而，PCA对噪声极为敏感，以致降低了其鲁棒性。与鲁棒PCA低秩分解算法不同，鲁棒PCA低维表征算法力求在不去噪的情形下提升鲁棒性。本文以目标函数和距离度量方式作为切入点，对当前主要的鲁棒PCA低维表征算法展开分析。首先，基于数据样本的处理形式、目标函数和距离度量准则等，对鲁棒PCA低维表征算法予以基本阐述。其次，按照目标函数的距离度量方式，深入剖析了一阶到高阶PCA的诸多典型算法，揭示了距离度量方式对PCA的特征提取、重构误差等性能的影响。最后，对四个国际标准数据集进行实验分析，在不同噪声条件下验证了典型PCA低维表征算法的鲁棒性。

Abstract:

Principal component analysis (PCA) is a classical data analysis and data dimensionality reduction algorithm that has been extensively used in several fields, including image compression and feature extraction. Nevertheless, PCA is extremely susceptible to noise, which compromises its robustness. In contrast to the robust low-rank decomposition algorithms, which are designed to remove noise from images, the robust low-dimensional representation algorithms seek to enhance the robustness without denoising. This paper starts with the objective function and distance metric, and analyzes the current main robust PCA low-dimensional representation algorithms. Firstly, the robust PCA low-dimensional representation algorithms are fundamentally elaborated according to the processing form of data samples, the optimization approach of the objective function and the distance metrics. Secondly, based on the distance metrics of the objective function, numerous typical algorithms from first-order to higher-order PCA were deeply analyzed, uncovering the influence of the distance metrics on the performance of PCA, such as feature extraction and reconstruction error. Finally, the robustness of typical PCA low-dimensional representation algorithm is validated under different noise conditions. This verification is carried out through empirical investigation utilizing the four international standard datasets.