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

תקציר מקורי באנגליתarXiv:2610.08069v1 Announce Type: new Abstract: 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 $J\sim\mathrm{Pois}(\kappa/2)$, where the budget allows deleting $k$ directions and $\kappa$ is the calibration signal-to-noise ratio. Projecting out the mean calibration difference attains it without knowing $\kappa$ or the noise scale. Thi