发表机构
Mohamed bin Zayed University of Artificial Intelligence (MBZUAI)（穆罕默德·本·扎耶德人工智能大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对数据源间均值偏移的移除，提出线性表示修复的统计决策框架，推导出精确极小极大风险，并揭示检测与修复所需信噪比的差距，为修正策略提供理论指导。
AI 中文摘要
两个数据源之间的均值偏移可能容易检测，但若不大幅改变其表示则难以移除。我们将其移除视为一个统计决策问题：从$\mathbb{R}^d$中配对校准测量的含噪差异中，学习一个线性映射，在硬失真预算下应用于两个源，使得在新数据上尽可能少地留下偏移。我们推导出所有此类映射上的精确有限样本极小极大风险，即$(d-k) \mathbb{E}[1/(d+2J)]$，其中$J\sim\mathrm{Pois}(\kappa/2)$，预算允许删除$k$个方向，$\kappa$是校准信噪比。投影出平均校准差异即可达到该风险，无需知道$\kappa$或噪声尺度。这揭示了检测-修复差距：检测偏移仅需$\kappa\gg\sqrt d$，而在恒定失真下移除其固定比例需要$\kappa\asymp d$，如同估计其方向。标准线性概念擦除器（MP、SAL、LEACE）移除相同的校准差异，因此该公式在拟合前精确给出它们在新数据上留下多少偏移以及目标需要多少校准。该极限是稳健的：配对使非高斯共享内容下仍精确，投影在异方差噪声下保持其保证，选择性弃权（不执行）无法缩小差距。在配对的临床和可穿戴睡眠脑电图上，参与者间差异作为校准噪声，该公式预测新参与者中留下的设备偏移，且每人更多记录很快不再有帮助。总之，这些结果说明一个不足的修正是否需要更好的方法、更多记录或更多参与者。
英文摘要
A mean shift between two data sources can be easy to detect but hard to remove without substantially changing their representations. We cast its removal as a statistical decision problem: from noisy differences between paired calibration measurements in $\mathbb{R}^d$, learn one linear map, applied to both sources under a hard distortion budget, that leaves as little of the shift as possible on fresh data. We derive the exact finite-sample minimax risk over all such maps, $(d-k) \mathbb{E}[1/(d+2J)]$ with $J\sim\mathrm{Pois}(κ/2)$, where the budget allows deleting $k$ directions and $κ$ is the calibration signal-to-noise ratio. Projecting out the mean calibration difference attains it without knowing $κ$ or the noise scale. This exposes a detection-repair gap: detecting the shift needs only $κ\gg\sqrt d$, whereas removing a fixed fraction of it at constant distortion needs $κ\asymp d$, as for estimating its direction. Standard linear concept erasers (MP, SAL, LEACE) remove the same calibration difference, so the formula gives, before fitting, exactly how much shift they leave on fresh data and how much calibration a target requires. The limit is robust: pairing keeps it exact for non-Gaussian shared content, the projection keeps its guarantee under anisotropic noise, and selective abstention cannot close the gap. On paired clinical and wearable sleep EEG, where differences between participants act as calibration noise, the formula predicts the device shift left in new participants, and more recordings per person soon stop helping. Together, these results tell whether a correction that falls short needs a better method, more recordings, or more participants.
Comments29 pages, 7 figures, 4 tables. Code: https://github.com/sneddy/shift-repair-frontier
