Detecting a Shift Is Not Enough: Exact Minimax Limits of Linear Representation Repair
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 ℝd, 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) 𝔼[1/(d+2J)] with J∼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≫√d, whereas removing a fixed fraction of it at constant distortion needs κasymp 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.
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BibTeX
@misc{aimoldin2026detecting,
title = {{Detecting a Shift Is Not Enough: Exact Minimax Limits of Linear Representation Repair}},
author = {Anuar Aimoldin and Yankai Chen and Ayana Mussabayeva and Xue Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08069},
eprint = {2610.08069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08069},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Detecting a Shift Is Not Enough: Exact Minimax Limits of Linear Representation Repair
AU - Anuar Aimoldin
AU - Yankai Chen
AU - Ayana Mussabayeva
AU - Xue Liu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08069
UR - https://arxiv.org/abs/2610.08069
AB - 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 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) 𝔼[1/(d+2J)] with J∼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≫√d, whereas removing a fixed fraction of it at constant distortion needs κasymp 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.
ER -