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主成分分析（Principal Component Analysis）

PublishedJuly 24, 2026Document8 pages

Contents

动机：PCA vs 随机投影 (Motivation: PCA vs Random Projection)
预处理 (Preprocessing)
PCA目标函数 (The PCA Objective)
寻找第一个成分 (Finding the First Component)
勾股定理论证 (Pythagorean Theorem Argument)
几何直觉：方差 = 分散度 (Geometric Intuition: Variance = Spread)
K个成分：PCA目标 (K Components: PCA Objective)
PCA后的表示 (Representation After PCA)
应用 (Applications)
应用1：基因地理学 (Genome Geography)
应用2：特征脸 (Eigenfaces)
协方差矩阵 (Covariance Matrix)
谱定理 (Spectral Theorem)
最大化二次型 (Maximizing the Quadratic Form)
简单情况：A为对角阵 (Simple Case: A is Diagonal)
一般情况：A = QDQT (General Case)
证明：XTX的特征值0 (Proof: Eigenvalues of XTX 0)
PCA算法总结 (PCA Algorithm Recap)
计算分解 (Computing the Decomposition)

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