Subspace Recovery from Repeated Binary Rules

We study recovery of a shared latent subspace from repeated binary classification tasks whose unknown rule vectors lie in that subspace. The learner observes Gaussian inputs and their binary labels, but not the rules themselves. With one observation per independently drawn rule, the subspace is unidentifiable. With two observations, we derive the exact likelihood and Fisher information. For fixed ambient dimension and known rank 3 ≤ k < d, we determine the asymptotic minimax constant under normalized squared projector loss and give an estimator that attains it. At rank three, a raw likelihood score can have infinite variance near an incorrect pilot, which requires a regularized correction. We also study a separate synthetic transfer setting with unmarked task boundaries. Retaining temporal spectral information improves early performance on independently drawn target tasks over erased history. This experiment assumes known rank and long histories and does not constitute an empirical verification of the minimax theorem.