Preconditioned Implicit Midpoint Langevin Sampling for Non-Smooth Bayesian Imaging

We introduce P-IMLA, a preconditioned implicit-midpoint Langevin sampler for high-dimensional log-concave targets with non-smooth regularization. The obstruction is that the large steps enabled by matrix preconditioning are precisely where preconditioned Euler--Maruyama inflates stationary variance. Implicit midpoint removes this surplus via a Cayley-transform identity, making P-IMLA Gaussian-exact at any step size. Beyond Gaussians, a Möbius-monotonicity argument gives an $M$-Wasserstein rate governed by $\kappa_{\rm eff}$ rather than the $\kappa_f$ of unpreconditioned IMLA, while backward-error analysis shows drift-only rather than Euler-type diffusion bias. An $M$-norm Moreau envelope yields a proximal non-smooth variant with a three-term error budget. Controlled Gaussian and TV-regularized imaging experiments confirm the predicted separation: the unpreconditioned IMLA degenerates as $\kappa_f$ grows, whereas P-IMLA remains $\kappa_{\rm eff}$-governed and near variance-exact.