[Submitted on 1 Nov 2026]
Abstract: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, whitening transfers the sharp IMLA contraction rate to $kappa_{mathrm{eff}}$, and backward-error analysis gives drift-only rather than Euler-type diffusion bias under explicit smoothness assumptions. 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 midpoint scheme retains long transients as $kappa_f$ grows, whereas P-IMLA remains $kappa_{mathrm{eff}}$-governed and near variance-exact.
Submission history
From: Zhipeng Lu [view email]
[v1] Sun, 1 Nov 2026 11:14:39 UTC (2,324 KB)
