When Does Auxiliary Calibration Improve Target Recovery? Nuisance Geometry, Sample Complexity, and Timing

Many learning problems combine target-bearing observations with auxiliary measurements that do not observe the target directly, but instead reveal a shared nuisance variable. When, and by how much, can such auxiliary measurements improve target recovery? We study this question in a Gaussian inverse experiment with an acquisition channel that mixes target and nuisance effects and a calibration channel that observes only the nuisance. We show that calibration value is determined not by marginal channel quality, but by the paired geometry relating the nuisance scores in the two channels. The closed relation pairing the two nuisance scores yields an exact resolvent formula for efficient target information, including nonclosed score ranges, and separates calibration-irrecoverable structural loss from finite-budget calibration loss. For unrestricted nuisance, we prove an exact minimax-risk reduction to the efficient Gaussian shift over compact convex target classes; for restricted nuisance classes, we give a nuisance-translation condition under which the reduction remains valid. Under a Hilbert-scale comparison, we derive matching two-resource minimax rates and the critical auxiliary sample size $m_{\mathrm{crit}}(n)$ needed to recover the nuisance-free target rate. In parabolic evolution models, calibration timing can change the effective ill-posedness and turn a polynomial auxiliary requirement into an exponential one, while nuisance constraints can change the minimax rate itself. The same geometry also yields target-oriented calibration allocation, which can differ from nuisance-optimal allocation. Experiments support these predictions and illustrate the additional errors-in-variables difficulty of learning the nuisance response.