A1883
Title: Poisson geometry of selective inference in proportional dimensions
Authors: Hirofumi Ota - University of Tokyo (Japan) [presenting]
Abstract: Conditional selective inference for the randomized Lasso yields exact finite-sample pivots, but exact validity does not guarantee informative confidence intervals in proportional dimensions. A Poisson-geometric theory is developed that explains the loss of post-selection precision caused by inactive KKT constraints. Under Gaussian design, an exact finite-sample factorization holds: conditional on the active residual, inactive scores are independent Gaussian variables. Consequently, near-binding inactive faces converge, after rescaling, to a marked Poisson point process, and the nearest inactive faces imply that the inactive truncation width is typically of order $\sqrt{n}$. The inactive selection event contributes an exact scalar post-selection tilt, reducing the analysis to a one-dimensional residual-norm state. In a downstream-small regime this state converges to a deterministic limit; in a proportional-active regime it is confined to a deterministic compact set. A nonasymptotic lower bound for truncated-normal confidence intervals converts the Poisson width scaling into $\sqrt{n}$ scale worst-case interval inflation.
