通过引入对抗性均值扰动，有效剔除主成分分析中的偏差成分。
Mean-Shift PCA by Knockoff Mean
- 设计敲击均值扰动，使噪声成分在谱上可分离
- 即使仅1%样本受偏移污染，主成分仍保持稳定
- 算法仅需标准PCA操作，适合高维数据处理
去除噪声困难，但添加噪声却很简单。本文提出通过刻意引入敲击均值扰动，消除主成分分析（PCA）中的均值漂移噪声成分。标准PCA对样本均值偏移极为敏感：少量来自偏移分布的样本即可导致主导主成分大幅偏离。在高维情形下，现有鲁棒PCA方法无法处理混合模型中固有的均值漂移污染结构。借助随机矩阵理论，我们证明均值漂移引起的特征值尖峰与原始协方差的稳定特征值在谱上可分离。此外，原始特征空间在渐近意义下不受污染影响，且独立于混合权重。基于此谱稳定性，我们提出一种简单的两阶段PCA算法：通过添加敲击均值，仅用标准PCA操作即可识别并移除均值漂移分量。
原文摘要 · Abstract (English)
Removing noise is difficult, but adding noise is easy. In this work, we show how to eliminate mean-shift noisy components from PCA by deliberately introducing knockoff mean-shift perturbation. Standard PCA is highly sensitive to shifts in the sample mean: a small fraction of samples from a shifted distribution can cause large deviations in the leading principal components. In high-dimensional regimes, existing Robust PCA approaches cannot handle the mean-shift contamination structure inherent in the mixture model. Using tools from Random Matrix Theory, we prove that the mean-shift spikes are spectrally separable from the stable eigenvalues of the original covariance. Furthermore, the original eigenspace remains asymptotically invariant to the contamination, independent of the mixture weight. Exploiting this spectral stability, we propose a simple, two-stage PCA algorithm by adding knockoff mean that identifies and removes the mean-shift component using only standard PCA operations.
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