[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119343-en":3,"doc-seo-119343-105":30,"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":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},119343,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees","Gaussian processes often serve in geospatial modeling, Bayesian optimization, and latent Gaussian models, where dependable numerical behavior is essential for correct downstream interaction. This work analyzes numerical stability of scalable sparse Gaussian process approximations based on inducing points by leveraging stability theory from interpolation. It derives sufficient—and in some cases necessary—conditions on inducing point placement to ensure stable computations. For low-dimensional tasks, it introduces an automated inducing point selection method using a modified cover tree, plus a stability-oriented sparse regression alternative.","Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees  \nAlexander Terenin􀀃  \nUniversity of Cambridge and Imperial College London David R. Burt􀀃  \nUniversity of Cambridge and MIT Artem Artemev􀀃  \nImperial College London and Secondmind  \nSeth Flaxman  \nUniversity of Oxford  \nMark van der Wilk  \nImperial College London and University of Oxford  \nCarl Edward Rasmussen  \nUniversity of Cambridge and Secondmind  \nHong Ge  \nUniversity of Cambridge  \nEditor: Mohammad Emtiyaz Khan  \nAbstract  \nGaussian processes are frequently deployed as part of larger machine learning and decision-making systems, for instance in geospatial modeling, Bayesian optimization, or in latent Gaussian models.  \nWithin a system, the Gaussian process model needs to perform in a stable and reliable manner to ensure it interacts correctly with other parts of the system. In this work, we study the numerical stability of scalable sparse approximations based on inducing points. To do so, we first review numerical stability, and illustrate typical situations in which Gaussian process models can be unstable. Building on stability theory originally developed in the interpolation literature, we derivesufficient and in certain cases necessary conditions on the inducing points for the computations performed to be numerically stable. For low-dimensional tasks such as geospatial modeling, we propose an automated method for computing inducing points satisfying these conditions. This is done via a modification of the cover tree data structure, which is of independent interest. We additionally propose an alternative sparse approximation for regression with a Gaussian likelihood which trades off a small amount of performance to further improve stability. We provide illustrative examples showing the relationship between stability of calculations and predictive performance of inducing point methods on spatial tasks.  \n1. Introduction  \nGaussian processes are a flexible framework and model class for learning unknown functions. Byway of being constructed in the language of Bayesian learning, Gaussian process models provide an ability to incorporate prior information into the model, and assess and propagate uncertainty in a principled manner. This makes them well-suited for a wide variety of areas where these capabilities  \n􀀃 Equal contribution.  \nCode available at: [https://github.com/awav/conjugate-gradient-sparse-gp](https://github.com/awav/conjugate-gradient-sparse-gp) .  \n􀀍c2024 Alexander Terenin, David R. Burt, Artem Artemev, Seth Flaxman, Mark van der Wilk, Carl Edward Rasmussen, and Hong Ge.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/. Attribution)[. Attribution](https://creativecommons.org/licenses/by/4.0/. Attribution) requirements are provided at  \n[http://jmlr.org/papers/v25/22-1170.html](http://jmlr.org/papers/v25/22-1170.html) .  \nTerenin, Burt, Artemev, Flaxman, van der Wilk, Rasmussen, and Ge  \nare important, including statistical applications such as spatial modeling (Cressie, 1992), and decisionmaking applications such as Bayesian optimization (Snoek et al., 2012), sensor placement (Krause et al., 2008), and active learning (Krause and Guestrin, 2007) .  \nIn many settings, the increased availability of data and need to accurately model higher-resolution phenomena has led to a strong interest in working with Gaussian processes at a larger scale. Unfortunately, classical Gaussian process models generally scale cubically with training data size due to the need to solve large linear systems of equations. This mismatch has led to a longstanding and fruitful line of work on scalable Gaussian processes. In the era of GPUs and automatic differentiation, two classes of scalable approximations have been deployed within major Gaussian process software packages, including GPflow (Matthews et al., 2017) and GPyTorch (Gardner et al., 2018): those based on inducing point methods (Qui","cbCaibg6sdP2DuC4","https://ap.wps.com/l/cbCaibg6sdP2DuC4","pdf",2590550,1,36,"English","en",105,"# Abstract\n# Introduction\n## Gaussian process scalability challenges\n## Inducing point and iterative approximation methods\n## Problem motivation and scope\n## Related work on numerical stability\n## Contributions and approach","[{\"question\":\"Why is numerical stability important for sparse Gaussian process approximations?\",\"answer\":\"Gaussian process models are used inside larger systems where unstable numerical computations can break reliability. The paper focuses on inducing-point sparse approximations to ensure computations remain stable across different datasets.\"},{\"question\":\"What conditions does the paper derive for inducing points to guarantee numerical stability?\",\"answer\":\"Using tools from interpolation-based stability theory, the paper derives sufficient conditions on inducing point placement and, for some cases, necessary conditions that ensure stable linear-algebra computations.\"},{\"question\":\"How does the proposed method compute inducing points for low-dimensional tasks?\",\"answer\":\"It proposes an automated inducing point selection procedure based on a modified cover tree data structure so that the selected inducing points satisfy the stability conditions.\"}]","Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees | PDF",1785723802,91,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"numerically-stable-sparse-gaussian-processes-via-minimum-separation-using-cover-trees","",{"@graph":36,"@context":85},[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/numerically-stable-sparse-gaussian-processes-via-minimum-separation-using-cover-trees/119343/",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","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is numerical stability important for sparse Gaussian process approximations?","Question",{"text":75,"@type":76},"Gaussian process models are used inside larger systems where unstable numerical computations can break reliability. The paper focuses on inducing-point sparse approximations to ensure computations remain stable across different datasets.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What conditions does the paper derive for inducing points to guarantee numerical stability?",{"text":80,"@type":76},"Using tools from interpolation-based stability theory, the paper derives sufficient conditions on inducing point placement and, for some cases, necessary conditions that ensure stable linear-algebra computations.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed method compute inducing points for low-dimensional tasks?",{"text":84,"@type":76},"It proposes an automated inducing point selection procedure based on a modified cover tree data structure so that the selected inducing points satisfy the stability conditions.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]