[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117582-en":3,"doc-seo-117582-105":30,"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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},117582,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems","Score-based diffusion models (SDMs) are used to sample from the posterior distribution in Bayesian inverse problems, but common approaches often require repeated evaluations of the forward operator per sample, making them costly for large-scale settings. This work introduces UCoS (unconditional representation of the conditional score) for linear inverse problems, moving computation to an offline training stage. The method learns a task-dependent score from the linear forward operator and derives the conditional score exactly from a trained unconditional score via affine transformations. The formulation is in infinite-dimensional function spaces and supports discretization-invariant convergence, validated on high-dimensional CT and deblurring experiments.","An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems  \nFabian Schneider  \nSchool of Engineering Science Lappeenranta-Lahti University of Technology Vienna University of Technology (TU Wien)  \nDuc-Lam Duong  \nSchool of Engineering Science  \nLappeenranta-Lahti University of Technology  \nMatti Lassas  \nDepartment of Mathematics and Statistics University of Helsinki  \nMaarten V. de Hoop  \nDepartment of Computational and Applied Mathematics Rice University  \nTapio Helin  \nSchool of Engineering Science  \nLappeenranta-Lahti University of Technology  \n[fabian.schneider@lut.fi](fabian.schneider@lut.fi)  \n[duc-lam.duong@lut.fi](duc-lam.duong@lut.fi)  \n[matti. lassas@helsinki.fi](matti. lassas@helsinki.fi)  \n[mdehoop@rice. edu](mdehoop@rice. edu)  \n[tapio.helin@lut.fi](tapio.helin@lut.fi)  \nReviewed on OpenReview: [https: // openreview. net/ forum? id= rO8erhXHPo](https: // openreview. net/ forum? id= rO8erhXHPo)  \nAbstract  \nScore-based diffusion models (SDMs) have emerged as a powerful tool for sampling from the posterior distribution in Bayesian inverse problems. However, existing methods often require multiple evaluations of the forward mapping to generate a single sample, resulting insignificant computational costs for large-scale inverse problems. To address this, we propose an unconditional representation of the conditional score function (UCoS) tailored to linear inverse problems, which avoids forward model evaluations during sampling by shifting computational effort to an offline training phase. In this phase, a task-dependent score function is learned based on the linear forward operator. Crucially, we show that the conditional score can be derived exactly from a trained (unconditional) score using affine transformations, eliminating the need for conditional score approximations. Our approach is formulated in infinite-dimensional function spaces, making it inherently discretization-invariant. We support this formulation with a rigorous convergence analysis that justifies UCoS beyond any specific discretization. Finally we validate UCoS through high-dimensional computed tomography (CT) and image deblurring experiments, demonstrating both scalability and accuracy.  \n1 Introduction  \nInverse problems seek to determine unknown quantities through indirect and noisy measurements, typically leading to ill-posed scenarios. The Bayesian approach to inverse problems frames the task as a quest for information. Blending statistical prior information of the unknown with a likelihood model for the measurement data gives rise to a posterior distribution, which fully characterizes the unknown conditioned on noisy data Kaipio & Somersalo (2006); Stuart (2010) . In severely ill-posed problems, the quality of inference  \nis strongly dependent on the expressivity of the prior. Traditional hand-crafted priors, such as the totalvariation prior, tend not to be expressive enough to characterize complicated structures Sun et al. (2023) . Generative models offer a flexible and computationally feasible approach to prior modeling as they offer the possibility of generating new samples after training on a data set characterizing the prior.  \nThis work investigates sampling from the posterior distribution of linear inverse problems using score-based diffusion models (SDMs) Song et al. (2021), which have recently received wide attention in the literature (in the context of inverse problems, see e.g. Batzolis et al. (2021); Lim et al. (2025); Hagemann et al. (2025); Graikos et al. (2022); Feng et al. (2023); Sun et al. (2023); Pidstrigach et al. (2024); Dey et al. (2024); Holzschuh et al. (2023); Dou & Song (2023); Barbano et al. (2025); Cardoso et al. (2023); Feng & Bouman (2023); Song et al. (2024); Meng & Kabashima (2022); Kveton et al. (2024); Wu et al. (2024a;b); Baldassari et al. (2024b); Yao et al. (2025); Chen et al. (2025)) . An SDM consists of two main components: a forward diffusion process and a ","cbCaicvfAIxXow7l","https://ap.wps.com/l/cbCaicvfAIxXow7l","pdf",12050359,1,37,"English","en",105,"# Introduction\n## Bayesian framing of inverse problems\n## Score-based diffusion models for posterior sampling\n## Conditional score estimation challenge\n## Proposed unconditional representation (UCoS)","[{\"question\":\"How is the conditional score obtained without conditional score approximations?\",\"answer\":\"The method is formulated in infinite-dimensional function spaces and includes rigorous convergence analysis that justifies UCoS beyond any specific discretization. It also supports scalability and accuracy in experiments like high-dimensional CT and image deblurring.\"}]","An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems | PDF",1785677096,93,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"an-unconditional-representation-of-the-conditional-score-in-infinite-dimensional-linear-inverse-problems","",{"@graph":36,"@context":77},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/an-unconditional-representation-of-the-conditional-score-in-infinite-dimensional-linear-inverse-problems/117582/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How is the conditional score obtained without conditional score approximations?","Question",{"text":75,"@type":76},"The method is formulated in infinite-dimensional function spaces and includes rigorous convergence analysis that justifies UCoS beyond any specific discretization. It also supports scalability and accuracy in experiments like high-dimensional CT and image deblurring.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":81,"canonical":52},"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":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]