Kalman-Langevin dynamics : exponential convergence, particle approximation and numerical approximation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909593995575296 |
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| author | Ringh, Axel Sharma, Akash |
| author_facet | Ringh, Axel Sharma, Akash |
| contents | Langevin dynamics has found a large number of applications in sampling, optimization and estimation. Preconditioning the gradient in the dynamics with the covariance - an idea that originated in literature related to solving estimation and inverse problems using Kalman techniques - results in a mean-field (McKean-Vlasov) SDE. We demonstrate exponential convergence of the time marginal law of the mean-field SDE to the Gibbs measure with non-Gaussian potentials. This extends previous results, obtained in the Gaussian setting, to a broader class of potential functions. We also establish uniform in time bounds on all moments and convergence in $p$-Wasserstein distance. Furthermore, we show convergence of a weak particle approximation, that avoids computing the square root of the empirical covariance matrix, to the mean-field limit. Finally, we prove that an explicit numerical scheme for approximating the particle dynamics converges, uniformly in number of particles, to its continuous-time limit, addressing non-global Lipschitzness in the measure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18139 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kalman-Langevin dynamics : exponential convergence, particle approximation and numerical approximation Ringh, Axel Sharma, Akash Probability Numerical Analysis Statistics Theory 65C30, 60H35, 60H10, 37H10, 35Q84 Langevin dynamics has found a large number of applications in sampling, optimization and estimation. Preconditioning the gradient in the dynamics with the covariance - an idea that originated in literature related to solving estimation and inverse problems using Kalman techniques - results in a mean-field (McKean-Vlasov) SDE. We demonstrate exponential convergence of the time marginal law of the mean-field SDE to the Gibbs measure with non-Gaussian potentials. This extends previous results, obtained in the Gaussian setting, to a broader class of potential functions. We also establish uniform in time bounds on all moments and convergence in $p$-Wasserstein distance. Furthermore, we show convergence of a weak particle approximation, that avoids computing the square root of the empirical covariance matrix, to the mean-field limit. Finally, we prove that an explicit numerical scheme for approximating the particle dynamics converges, uniformly in number of particles, to its continuous-time limit, addressing non-global Lipschitzness in the measure. |
| title | Kalman-Langevin dynamics : exponential convergence, particle approximation and numerical approximation |
| topic | Probability Numerical Analysis Statistics Theory 65C30, 60H35, 60H10, 37H10, 35Q84 |
| url | https://arxiv.org/abs/2504.18139 |