{"paper":{"title":"Dimension-Uniform Discretization Analysis of Preconditioned Annealed Langevin Dynamics for Multimodal Gaussian Mixtures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Exponential-integrator discretization of preconditioned annealed Langevin dynamics yields dimension-uniform KL bounds for Gaussian mixtures under spectral summability conditions.","cross_cats":["cs.LG","cs.NA","math.NA","math.PR"],"primary_cat":"stat.ML","authors_text":"Josselin Garnier, Knut Solna, Lorenzo Baldassari, Maarten V. de Hoop","submitted_at":"2026-05-15T15:18:32Z","abstract_excerpt":"Obtaining stable diffusion-based samplers in high- and infinite-dimensional settings is challenging because errors can accumulate across high-frequency coordinates and make the dynamics unstable under refinement of the finite-dimensional approximation of the underlying function-space problem. Discretization is a typical source of such errors, and preconditioning with a suitable spectral decay is one way to control their accumulation. In this paper, we study this problem for preconditioned annealed Langevin dynamics (ALD) applied to Gaussian mixtures. We first show that Euler-Maruyama (EM) disc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Under explicit spectral summability conditions coupling the smoothing covariance, the component covariance spectra, and the preconditioner, we prove a dimension-uniform Kullback-Leibler (KL) bound for this scheme. This bound can be made arbitrarily small, uniformly in dimension, by allowing enough time for annealing and then refining the time mesh accordingly. Importantly, these conditions allow regimes in which the KL divergence between the target and the initial smoothed law diverges with dimension.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The target is a finite Gaussian mixture whose component covariance spectra satisfy the stated summability conditions together with the chosen smoothing covariance and preconditioner (abstract, paragraph on exponential-integrator scheme).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves dimension-uniform KL bounds for exponential-integrator discretization of preconditioned ALD on Gaussian mixtures under spectral summability, showing EM stability restrictions are scheme-dependent rather than intrinsic.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Exponential-integrator discretization of preconditioned annealed Langevin dynamics yields dimension-uniform KL bounds for Gaussian mixtures under spectral summability conditions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"30eab76c74f57e831e70c868c2a9fc6853fd3d5777e96646bef97c8d9c2f53fa"},"source":{"id":"2605.16473","kind":"arxiv","version":1},"verdict":{"id":"0f15c4d4-e97b-4680-9c7e-215a72f72435","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:55:23.428361Z","strongest_claim":"Under explicit spectral summability conditions coupling the smoothing covariance, the component covariance spectra, and the preconditioner, we prove a dimension-uniform Kullback-Leibler (KL) bound for this scheme. This bound can be made arbitrarily small, uniformly in dimension, by allowing enough time for annealing and then refining the time mesh accordingly. Importantly, these conditions allow regimes in which the KL divergence between the target and the initial smoothed law diverges with dimension.","one_line_summary":"Proves dimension-uniform KL bounds for exponential-integrator discretization of preconditioned ALD on Gaussian mixtures under spectral summability, showing EM stability restrictions are scheme-dependent rather than intrinsic.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The target is a finite Gaussian mixture whose component covariance spectra satisfy the stated summability conditions together with the chosen smoothing covariance and preconditioner (abstract, paragraph on exponential-integrator scheme).","pith_extraction_headline":"Exponential-integrator discretization of preconditioned annealed Langevin dynamics yields dimension-uniform KL bounds for Gaussian mixtures under spectral summability conditions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16473/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T22:01:30.236484Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:23.272977Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:33:23.115506Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:21:57.048513Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"cbd0694bb6ea4d416b20ce88a6c1a180d0a61b84a0f660b3b8ec4f10c86cdf6e"},"references":{"count":54,"sample":[{"doi":"","year":2025,"title":"Lorenzo Baldassari, Josselin Garnier, Knut Sølna, and Maarten V . de Hoop. Preconditioned Langevin dynamics with score-based generative models for infinite-dimensional linear Bayesian inverse problems","work_id":"a4532702-7d3b-4002-8a18-5590d0b930f8","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Dimension-free multimodal sampling via preconditioned annealed langevin dynamics","work_id":"1e3ae0e3-c5d1-4165-9f86-6a5d092e0c64","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Lorenzo Baldassari, Ali Siahkoohi, Josselin Garnier, Knut Sølna, and Maarten V . de Hoop. Conditional score-based diffusion models for Bayesian inference in infinite dimensions. In Proceedings of the ","work_id":"7c6f1a46-247b-411b-9465-001b93122e9c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"year = 2024, month = may, number =","work_id":"56d90e12-729e-4653-8976-57ec743ede53","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"Geometric MCMC for infinite-dimensional inverse problems.Journal of Computational Physics, 335:327–351, 2017","work_id":"e1b8193b-3930-4836-b3b0-9ca30532923f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":54,"snapshot_sha256":"8161254eace7428bfa0fca7c01df3cc439d3a872f182c55721844bd067c6a9f4","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}