Preconditioned Crank–Nicolson Langevin sampling
Algorithms AIMech:PreconditionedCrankNicolsonLangevin DRAFT
Aliases: pCNL
A Metropolis-corrected Gaussian-reference sampler that preconditions the potential-gradient drift by the reference covariance and uses a scalar Crank–Nicolson autoregressive update.
Identity Notes & Concept Scope
This is Cotter et al. pCNL. The reference covariance multiplies grad Phi, and the linear denominator is the scalar 2+delta, distinguishing it from the CNL matrix solve. Removing the potential gradient recovers the pCN proposal, but a nonzero gradient changes its acceptance rule. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 167 prior native IDs/names/aliases. Full-checkout searches in both worktrees included ignored and hidden files; the 15 lines in each were shared Tierney source-title references or an adaptive-rejection discovery lead, not matching records. Existing Gibbs, Metropolis-within-Gibbs, random-walk and independence Metropolis mechanisms were inspected to resolve specialization boundaries.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
For finite-dimensional pi(u) proportional to exp(−Phi(u)) Normal(u;0,C), with fixed symmetric positive-definite C, assume differentiable Phi, choose delta>0 and draw w from Normal(0,C). Set v=((2−delta)u−2 delta C grad Phi(u)+sqrt(8 delta)w)/(2+delta). Its Gaussian covariance is 8 delta C/(2+delta) squared. Let q be that proposal; accept with min(1,pi(v)q(u|v)/(pi(u)q(v|u))) and otherwise retain u, using the reverse gradient-dependent mean at v. This finite-dimensional expression specializes the source’s general Metropolis rule and pCNL rho acceptance. The prior-cancelled pCN likelihood ratio alone is generally insufficient for this gradient-directed kernel.
⚠️ Curated Limitations & Negative Results
- The target must be proper relative to the Gaussian reference. The explicit density-ratio implementation here is finite-dimensional with symmetric positive-definite C; function-space extensions require the source’s operator, regularity and absolute-continuity assumptions.
- Target invariance does not imply independent draws, good exploration or finite-run accuracy. Mesh-robust behavior under stated assumptions is not a universal dimension-free performance guarantee.
- C and the step size are fixed in the stated kernel. Adaptive covariance or step-size policies need separate validity arguments.
- The reprint’s later Equation (6.2) reverses signs relative to Equation (6.1) and the explicit Section 4 proposals. This record uses Section 4 and does not claim a published erratum or use the inconsistent expression.
- The gradient is of the potential relative to the Gaussian reference. Substituting the full log-target gradient without accounting for the already implicit Gaussian drift defines a different proposal.
- Function-space validity and efficient discretizations need the source’s extra assumptions; the finite-dimensional density ratio is not a definition of an infinite-dimensional Lebesgue density.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | This algorithm constructs target-invariant Markov transitions under its stated assumptions. The exact AIO CLASS denotes a stochastic process/model and is exported beneath Network. This is a method-to-process related match, not equivalence, subclass membership, independence of retained draws or a claim that the algorithm is a trained network. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
S. L. Cotter, G. O. Roberts, A. M. Stuart, D. White
|
primary paper
full_text
|
ID: doi:10.1214/13-STS421
sha256:4be2044ef1d8…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T07:13:10Z
Updated: 2026-10-11T07:13:10Z