Detecting a Shift Is Not Enough: Exact Minimax Limits of Linear Representation Repair

Anuar Aimoldin, Yankai Chen, Ayana Mussabayeva, Xue Liu

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…