[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85510-en":3,"doc-seo-85510-105":29,"detail-sidebar-cat-0-en-105":83},{"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},85510,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Bures-Wasserstein Importance-Weighted Evidence Lower Bound Exposition and Applications","Importance-weighted evidence lower bound (IW-ELBO) is used for variational inference because it tightens the standard ELBO and reduces pathological mode-seeking. Yet Euclidean-space optimization becomes inefficient as the number of importance samples K grows, since Euclidean gradient estimators suffer a vanishing signal-to-noise ratio. This work reformulates IW-ELBO optimization in Bures-Wasserstein (BW) space for Gaussian VI by deriving and projecting the infinite-dimensional Wasserstein gradient. The resulting BW gradient attains favorable SNR scaling, avoids vanishing behavior, extends to a variational Rényi bound, and shows improved empirical performance.","arXiv :2602 .04272v2 [ stat .CO] 11 Jul 2026  \nBures-Wasserstein Importance-Weighted Evidence Lower Bound:  \nExposition and Applications ∗  \nPeiwen Jiang, Takuo Matsubara, and Minh-Ngoc Tran  \nThe University of Sydney Business School, Australia  \nAbstract  \nThe importance-weighted evidence lower bound (IW-ELBO) has emerged as a compelling objective for variational inference (VI), providing a tighter bound than the standard ELBO and mitigating its pathological mode-seeking behavior. However, optimizing the IW-ELBO in Euclidean space can be inefficient: as the number of importance samples K increases, standard Euclidean gradient estimators suffer from a vanishing signal-to-noise ratio (SNR) . This paper reformulates the optimization of the IW-ELBO within the Bures-Wasserstein (BW) space, the manifold of Gaussian distributions equipped with the 2-Wasserstein metric. We derive the infinitedimensional Wasserstein gradient of the IW-ELBO and project it onto the BW space, yielding a computationally tractable BW gradient for Gaussian VI. While the SNR of the Euclidean gradient is known to vanish at a rate of O(1/ √K  ) , we prove that the SNR of the Wasserstein gradient scales favorably at a rate of Ω( √K  ) . Crucially, the BW gradient maintains computational tractability while inheriting the stability of the Wasserstein geometry. We establish that its SNR interpolates between the Euclidean and Wasserstein regimes, achieving a non-vanishing rate of Ω(1) . We extend this geometric analysis to the variational Rényi importance-weighted autoencoder bound, proving analogous stability guarantees. Empirical evaluations demonstrate that the proposed algorithm achieves superior mass-covering and algorithmic performance compared to established baselines.  \nKeywords: Variational Inference, Wasserstein Space, Optimal Transport, Wasserstein Gradient Descent  \n1 Introduction  \nDriven by the growing sophistication of statistical models, posterior distributions in modern applications have increasingly become high-dimensional and complex. This computational burden has motivated a shift from classical sampling techniques toward scalable alternatives. Two prominent paradigms have emerged at the forefront of this transition: variational inference (VI) and Wasserstein gradient flow (WGF) . VI (Jordan et al., 1999) frames posterior approximation as an optimization problem over the parameter of a tractable density family, a task typically achieved by maximizing the evidence lower bound (ELBO) . Gaussian VI, for example, is widely employed to optimize the mean and covariance within a Gaussian family. In contrast, WGF (Jordan et al., 1998) offers a  \n∗ Acknowledgement: We are grateful to the Editor, Associate Editor and the three anonymous reviewers for their careful reading of the manuscript and for their constructive comments and suggestions, which have substantially improved the paper.  \nnon-parametric, geometric framework, under which the target-density approximation is formally characterized as an infinite-dimensional optimization over the space of probability distributions.  \nRecent literature has made substantial progress in unifying the two distinct paradigms of parametric VI and non-parametric WGF. Notably, Lambert et al. (2022) established that Gaussian VI can be recast as a WGF restricted to the manifold of Gaussian distributions. This unification relies on equipping the space of Gaussian measures with the 2-Wasserstein metric, giving rise to the Bures-Wasserstein (BW) space (Bures, 1969) . The BW geometry elegantly bridges the differential calculus in the parameter space with the infinite-dimensional optimization landscape over probability distributions. This geometric perspective facilitates novel theoretical insights and algorithmic developments by importing convex optimization techniques from optimal transport into VI. Leveraging this framework, Diao et al. (2023) developed an efficient algorithm for Gaussian VI that adapts a forward-backward ","cbCaiiB441ingkT0","https://ap.wps.com/l/cbCaiiB441ingkT0","pdf",2804821,1,43,"English","en",105,"# Introduction\n## Variational Inference and Wasserstein Gradient Flow\n## IW-ELBO and Its Limitations\n## Bridging IW-ELBO with BW Geometry\n# Wasserstein and BW Gradient","[{\"question\":\"How does the signal-to-noise ratio (SNR) of the BW gradient behave compared with the Euclidean gradient?\",\"answer\":\"The Euclidean gradient SNR vanishes at rate O(1/√K), while the Wasserstein/BW gradient SNR is proved to scale favorably, including a non-vanishing Ω(1) regime that interpolates between Euclidean and Wasserstein behavior.\"}]",1784204086,108,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":27},"bures-wasserstein-importance-weighted-evidence-lower-bound-exposition-and-applications","",{"@graph":35,"@context":77},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/bures-wasserstein-importance-weighted-evidence-lower-bound-exposition-and-applications/85510/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How does the signal-to-noise ratio (SNR) of the BW gradient behave compared with the Euclidean gradient?","Question",{"text":75,"@type":76},"The Euclidean gradient SNR vanishes at rate O(1/√K), while the Wasserstein/BW gradient SNR is proved to scale favorably, including a non-vanishing Ω(1) regime that interpolates between Euclidean and Wasserstein behavior.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]