Detecting a Shift Is Not Enough: Exact Minimax Limits of Linear Representation Repair
Exact minimax limits show linear shift repair needs far more calibration signal than detection.
The paper treats removal of a mean shift between two sources as learning one linear map under a hard distortion budget from noisy paired calibration measurements. It derives the exact finite-sample minimax risk, (d-k) times the expectation of 1/(d+2J) with J Poisson(kappa/2), and shows that projecting out the mean calibration difference attains it. Detecting the shift needs only kappa much larger than sqrt(d), while removing a fixed fraction at constant distortion needs kappa on the order of d. On paired clinical and wearable sleep EEG, the formula predicts residual device shift in new participants.
- Exact minimax risk is derived for budgeted linear shift repair.
- Detection needs kappa much larger than sqrt(d); fixed-fraction repair needs kappa on the order of d.
- Projecting out the mean calibration difference attains the bound without knowing kappa.
- The formula predicts residual shift left by MP, SAL, and LEACE.
- On paired sleep EEG, extra recordings per person soon stop helping.
From the original article
1 min read at arxiv.orgA 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.
By Anuar Aimoldin · arxiv.org
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