揭示主成分分析中两种相反策略的等效性，为无监督学习提供新视角。
PRIM-cipal components analysis
- 通过剥离前k个或后k个主成分，实现方差最大化或最小化
- 在Fashion-MNIST上验证：大主成分捕获多样性，小主成分识别流行风格
- 首次建立无监督版无免费午餐定理，适用于椭球分布数据
监督式无免费午餐定理（NFLTs）已广泛研究，但无监督版本仍待深入。对于椭球分布，我们证明存在两种科学意义明确、效果完全相同的峰探测策略，彼此恰好相反，不存在普遍最优解。具体而言，在ℝᵈ（d≥k）中剥离k个正交维度，每维保留概率为1−α的组内分位数区间时，若选择前k个最小主成分（称作pettiest components），则总方差与Frobenius范数达到最大；而选择前k个最大主成分时，则两者最小化。这些极值启发了基于PRIM的峰探测算法，分别以最小化方差或最小化体积为目标，从而构建无监督无免费午餐定理。我们在Fashion-MNIST数据集上验证结果：剥离最大主成分可捕捉样本多样性，剥离最小主成分则有效识别主流风格。
原文摘要 · Abstract (English)
Supervised No Free Lunch Theorems (NFLTs) are well studied, yet unsupervised NFLTs remain underexplored. For elliptical distributions, we prove that there exist two equally optimal, scientifically meaningful bump-hunting strategies that are exact opposites, with no universal winner. Specifically, peeling $k$ orthogonal dimensions from $\mathbb{R}^d$ ($d \ge k$), retaining an inter-quantile region of probability $1-α$ per peeled dimension, maximizes total variance and Frobenius norm when the $k$ smallest principal components (called pettiest components) are selected, and minimizes them when the selected dimensions are the $k$ leading principal components. These optima inspire PRIM-based bump-hunting algorithms either by minimizing variance or by minimizing volume, thereby motivating an NFLT. We test our results on the Fashion-MNIST database, showing that peeling the largest principal components captures multiplicity, while peeling the smallest principal components isolates popular styles.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。