[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85418-en":3,"doc-seo-85418-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85418,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Efficient sampling for sparse Bayesian learning using hierarchical prior normalization","Efficient Markov chain Monte Carlo (MCMC) sampling method for challenging high-dimensional sparse Bayesian learning (SBL) posteriors. The approach uses hierarchical prior-normalizing transport maps (TMs), deterministic couplings derived from the product-like structure of SBL priors and Knothe–Rosenblatt rearrangements. These maps transform the complex posterior into a simpler reference distribution with a standard normal prior, enabling structure-exploiting samplers. Numerical experiments on inverse problems such as signal deblurring, nonlinear inviscid Burgers inversion, and impulse image recovery show substantial gains over standard MCMC.","arXiv :2505 .23753v2 [math .NA] 11 Jul 2026  \nEfficient sampling for sparse Bayesian learning using hierarchical prior normalization ∗  \nJan Glaubitz† and Youssef M. Marzouk‡  \n\n| Abstract. We introduce an approach for efficient Markov chain Monte Carlo (MCMC) sampling for challenging highdimensional distributions in sparse Bayesian learning (SBL) . The core innovation involves using hierarchical prior-normalizing transport maps (TMs), which are deterministic couplings that transform the sparsitypromoting SBL prior into a standard normal one. We analytically derive these prior-normalizing TMs by leveraging the product-like form of SBL priors and Knothe–Rosenblatt (KR) rearrangements. These transform the complex target posterior into a simpler reference distribution equipped with a standard normal prior that can be sampled more efficiently. Specifically, one can leverage the standard normal prior by using more efficient, structure-exploiting samplers. Our numerical experiments on various inverse problems—including signal deblurring, inverting the non-linear inviscid Burgers equation, and recovering an impulse image—demonstrate significant performance improvements for standard MCMC techniques.\u003Cbr>Key words. Sparse Bayesian learning, inverse problems, hierarchical prior normalization, MCMC sampling AMS subject classifications. 62F15, 65C05, 65C40, 68U10\u003Cbr>Code repository. [https://github.com/jglaubitz/paper-2025-SBL-priorNormalization](https://github.com/jglaubitz/paper-2025-SBL-priorNormalization)\u003Cbr>DOI. [https://doi.org/10.1137/25M1790427](https://doi.org/10.1137/25M1790427) |\n| --- |\n| 1. Introduction. Many applications involve recovering an unknown high-dimensional parameter vector x ∈ Rn from noisy, indirect, and limited observational data y ∈ Rm by solving an inverse problem. For instance, for additive noise, one considers the data model\u003Cbr>(1.1) y = F (x) + e,\u003Cbr>where F : Rn → Rm is a known forward operator and e ∈ Rm denotes an unknown additive noise component. The Bayesian approach [71, 22] frames (1.1) as a statistical inference problem based on the posterior distribution π y , which combines the likelihood function f (x;y) implied by (1.1) with a prior density π 0 that encodes our structural beliefs about x.\u003Cbr>One can often assume that x is sparse or has a sparse representation—for instance, in the edge domain when x contains the nodal values of a piecewise smooth function. There are various sparsitypromoting priors, including Laplace [34, 9], Cauchy [49, 72], and horseshoe [25, 76 , 32] priors. Here, we consider the particularly potent class of hierarchical SBL priors π 0 (x, θ) = π 0 (x|θ)π 0 (θ) . These combine a conditionally Gaussian prior π0 (x|θ) with a generalized gamma hyper-prior π0 (θ) . SBL priorshave been demonstrated to provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. See [74, 21] for early versions of SBL based on (inverse) gamma hyper-priors, [20, 19] for their extension to generalized gamma hyper-priors, [24, 69] for their analysis, and [41, 82 , 40 , 46 , 47] for generalized SBL (GSBL) models for promoting linear transforms, Rx, to be sparse. The SBL posterior density πy for x, θ|y is given by Bayes’ theorem as\u003Cbr>(1.2) πy (x, θ) = 1Z f (x;y)π 0 (x, θ), |\n\n∗ Submitted to the editors July 14, 2026  \nFunding: JG and YM acknowledge support from the US Department of Defense, MURI program under grant number \\#N00014-20-1-2595 and from the US Department of Energy, SciDAC program (ASCR/HEP) under grant number \\#DESC0012704 . JG further acknowledges support by the Swedish Research Council (VR) Starting Grant \\#2025-05370, the Zenith Career Development Grant \\#26.07, and the National Academic Infrastructure for Supercomputing in Sweden (NAISS) grants \\#2025/22-1599 and \\#2024/22-1207 .  \n† Department of Mathematics, Link¨oping University, Sweden ([jan.glaubitz@liu.se](jan.glaubitz@liu.se))  \n‡ Department of Aeronautics and Astronautics, Massachusetts Institute of Tec","cbCaitW9hBmBbuQS","https://ap.wps.com/l/cbCaitW9hBmBbuQS","pdf",8104707,1,26,"English","en",105,"# Introduction\n## Sparse Bayesian learning and hierarchical priors\n## Challenges of MCMC for SBL posteriors","[{\"question\":\"What is the main idea behind the proposed efficient MCMC sampling method?\",\"answer\":\"The method constructs hierarchical prior-normalizing transport maps that deterministically transform the sparsity-promoting SBL prior into a standard normal reference prior, making sampling more efficient.\"},{\"question\":\"How are the transport maps derived in the approach?\",\"answer\":\"They are analytically derived by exploiting the product-like form of SBL priors together with Knothe–Rosenblatt (KR) rearrangements to build the prior-normalizing transformation.\"},{\"question\":\"Which inverse problems were used to validate the performance improvements?\",\"answer\":\"The experiments include signal deblurring, inversion of the nonlinear inviscid Burgers equation, and recovery of an impulse image, demonstrating improved performance for standard MCMC 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is the main idea behind the proposed efficient MCMC sampling method?","Question",{"text":75,"@type":76},"The method constructs hierarchical prior-normalizing transport maps that deterministically transform the sparsity-promoting SBL prior into a standard normal reference prior, making sampling more efficient.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the transport maps derived in the approach?",{"text":80,"@type":76},"They are analytically derived by exploiting the product-like form of SBL priors together with Knothe–Rosenblatt (KR) rearrangements to build the prior-normalizing transformation.",{"name":82,"@type":73,"acceptedAnswer":83},"Which inverse problems were used to validate the performance improvements?",{"text":84,"@type":76},"The experiments include signal deblurring, inversion of the nonlinear inviscid Burgers equation, and recovery of an impulse image, demonstrating improved performance for standard MCMC 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