对比PCA与SVD在降维中的表现，给出不依赖实验的选型建议。
A Comparative Analysis of Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) as Dimensionality Reduction Techniques
- 从原理出发推导两种方法，分析其数学本质。
- 评估解释性、数值稳定性和不同矩阵形状下的适用性。
- 提供无需实测即可选择算法的实用指南。
高维图像数据在进一步分析前通常需要降维。本文对两种线性降维技术——主成分分析（PCA）和奇异值分解（SVD）进行了纯解析比较。在从基本原理推导两种算法后，评估了它们的可解释性、数值稳定性以及对不同矩阵形状的适应性。基于经典与近期数值分析文献，归纳出无需实证基准测试即可选择其中一种算法的实用规则。最后讨论了方法局限性及未来实验研究方向。
原文摘要 · Abstract (English)
High-dimensional image data often require dimensionality reduction before further analysis. This paper provides a purely analytical comparison of two linear techniques-Principal Component Analysis (PCA) and Singular Value Decomposition (SVD). After the derivation of each algorithm from first principles, we assess their interpretability, numerical stability, and suitability for differing matrix shapes. We synthesize rule-of-thumb guidelines for choosing one out of the two algorithms without empirical benchmarking, building on classical and recent numerical literature. Limitations and directions for future experimental work are outlined at the end.
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