按组平衡重构误差，逐个构建公平主成分。
Fair PCA, One Component at a Time
- 逐个生成主成分，每步保证组间误差最小时的正交方向。
- 在多组数据上比现有方法更均衡地降低各组重构误差。
- 保留标准PCA的嵌套结构，适合需要公平性与可解释性的场景。
最小最大公平主成分分析（Min-Max Fair PCA）旨在为多组数据寻找低秩表示，使各组的近似误差尽可能均衡。现有方法虽能输出秩为d的公平子空间，但缺乏标准PCA的核心包含性质：即所有低秩子空间应被高秩子空间包含。为此，本文定义公平主成分：在与先前选成分正交的约束下，最小化组间最大重构误差的方向，并提出一种迭代算法计算这些成分。该方法保持了标准PCA的包含性质，且在单组数据时退化为标准PCA。理论分析表明其良好性质，实验证明其在最小最大公平性上优于现有方法。
原文摘要 · Abstract (English)
The Min-Max Fair PCA problem seeks a low-rank representation of multi-group data such that the the approximation error is as balanced as possible across groups. Existing approaches to this problem return a rank-$d$ fair subspace, but lack the fundamental containment property of standard PCA: each rank-$d$ PCA subspace should contain all lower-rank PCA subspaces. To fill this gap, we define fair principal components as directions that minimize the maximum group-wise reconstruction error, subject to orthogonality with previously selected components, and we introduce an iterative method to compute them. This approach preserves the containment property of standard PCA, and reduces to standard \pca for data with a single group. We analyze the theoretical properties of our method and show empirically that it outperforms existing approaches to Min-Max Fair PCA.
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