Preconditioned Implicit Midpoint Langevin Sampling for Non-Smooth Bayesian Imaging
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 κeff, 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 κf grows, whereas P-IMLA remains κeff-governed and near variance-exact.
Related
- Wasserstein Bounds for Euler and Randomized Midpoint Langevin Sampling with Noisy Gradients
- Sharp dimensional analysis of midpoint methods for Langevin sampling
- Evolutionary stochastic gradient Langevin dynamics: overcoming local traps in large-scale Bayesian sampling.
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- Strong convergence rate of the implicit midpoint method for stochastic KdV equations
BibTeX
@misc{wang2026preconditioned,
title = {{Preconditioned Implicit Midpoint Langevin Sampling for Non-Smooth Bayesian Imaging}},
author = {Sichen Wang and Zhipeng Lu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09980},
eprint = {2611.09980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09980},
}
RIS
TY - UNPB
TI - Preconditioned Implicit Midpoint Langevin Sampling for Non-Smooth Bayesian Imaging
AU - Sichen Wang
AU - Zhipeng Lu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.09980
UR - https://arxiv.org/abs/2611.09980
AB - 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 κ_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 κ_f grows, whereas P-IMLA remains κ_eff-governed and near variance-exact.
ER -