Standard Principal-Component Analysis (PCA) is known to be sensitive to outliers among the processed data. On the other hand, L1-norm-based PCA (L1-PCA) exhibits sturdy resistance against outliers, while it performs similar to standard PCA when applied to nominal or smoothly corrupted data [1]. Exact calculation of the $K$ L1-norm Principal Components (L1-PCs) of a rank-$r$ data matrix $\mathbf X \in \mathbb {R}^{D \times N}$ costs $\mathcal {O}(N^{(r -1)K + 1})$ [1], [2]. In this work, we present reduced-rank L1-PCA (RR L1-PCA): a hybrid approach that approximates the $K$ L1-PCs of $\mathbf X$ by the L1-PCs of its L2-norm-based rank-$d$ approximation ($d \leq r$), calculable exactly with reduced complexity $\mathcal {O}(N^{(d -1)K + 1})$. The proposed method combines the denoising capabilities and low computation cost of standard PCA with the outlier-resistance of L1-PCA. RR L1-PCA is accompanied by formal performance guarantees as well as thorough numerical studies that corroborate its computational and corruption resistance merits.

中文翻译：

---

具有性能保证的降阶 L1 范数主成分分析

众所周知，标准主成分分析 (PCA) 对处理数据中的异常值很敏感。另一方面，基于 L1 范数的 PCA (L1-PCA) 对异常值表现出强大的抵抗力，而当应用于标称或平滑损坏的数据时，它的表现类似于标准 PCA[1]. 精确计算$K$ 秩的 L1 范数主成分 (L1-PC)-$r$ 数据矩阵 $\mathbf X \in \mathbb {R}^{D \times N}$ 成本 $\mathcal {O}(N^{(r -1)K + 1})$ [1], [2]. 在这项工作中，我们提出了降阶 L1-PCA（RR L1-PCA）：一种近似于$K$ L1-PCs $\mathbf X$ 由其基于 L2 范数的等级的 L1-PC-$d$ 近似值（$d \leq r$)，可精确计算，降低复杂度 $\mathcal {O}(N^{(d -1)K + 1})$. 所提出的方法结合了标准 PCA 的去噪能力和低计算成本以及 L1-PCA 的抗异常值能力。RR L1-PCA 伴随着正式的性能保证以及彻底的数值研究，证实了其计算和抗腐败的优点。