Diffusion Bootstrap for High-Dimensional Linear Models

Classical bootstrap methods can behave poorly in high-dimensional linear models: the pairs bootstrap often yields overly conservative inference, whereas the residual bootstrap can be anti-conservative, reflecting systematic failures in variance calibration.

We propose a diffusion-based pairs bootstrap that replaces the empirical joint distribution with a learned generative law.

We establish variance consistency under a score approximation assumption, using complementary SDE and PDE arguments.

Counterexamples show that terminal $W_4$ convergence alone is insufficient for variance consistency.

Experiments indicate that diffusion pairs bootstrap improves variance calibration and generally improves Type~I error calibration, including in settings not covered by our theory.