[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124562-en":3,"doc-seo-124562-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},124562,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","ALGEBRAIC MACHINE LEARNING WITH AN APPLICATION TO CHEMISTRY","As scientific datasets grow in complexity, data analysis increasingly relies on the geometry and topology of data, including tools like persistent homology. Yet topological methods give only coarse information, while manifold-based approaches assume the data lies on a smooth manifold, which fails when the underlying space has singularities common in physical models. This paper presents an algebraic machine learning pipeline learning varieties without smoothness assumptions, using a MAP formulation solved via eigenvalue computation, then extracting geometric information with Gröbner bases and numerical methods, including a singularity-detection heuristic.","arXiv :2205 .05795v2 [math .AG] 13 May 2022  \nALGEBRAIC MACHINE LEARNING WITH AN APPLICATION TO  \nCHEMISTRY  \nEZZEDDINE EL SAI, PARKER GARA, AND MARKUS J. PFLAUM  \nAbstract . As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process.  \nThis can be seen for example with the growing interest in topological tools such as persistent homology. However, on the one hand, topological tools are inherently limited to providing only coarse information about the underlying space of the data. On the other hand, more geometric approaches rely predominately on the manifold hypothesis, which asserts that the underlying space is a smooth manifold. This assumption fails for many physical models where the underlying space contains singularities.  \nIn this paper we develop a machine learning pipeline that captures fine-grain geometric information without having to rely on any smoothness assumptions. Our approach involves working within the scope of algebraic geometry and algebraic varieties instead of differential geometry and smooth manifolds. In the setting of the variety hypothesis, the learning problem becomes to find the underlying variety using sample data. We cast this learning problem into a Maximum A Posteriori optimization problem which we solve in terms of an eigenvalue computation. Having found the underlying variety, we explore the use of Gröbner bases and numerical methods to reveal information about its geometry. In particular, we propose a heuristic for numerically detecting points lying near the singular locus of the underlying variety.  \n1. Introduction  \nExploring the geometry of data has shown to provide significant insight into high dimensional complex datasets. Applications include dimensionality reduction [22], computer vision [26], chemistry [21] and medicine [23] . Geometric methods in machine learning rely predominantly on the manifold hypothesis [13] which asserts that sample data 􀀊 􀀚 Rn in fact live in a smooth submanifold 􀀊 􀀚 M 􀀚 Rn whose dimension is often much smaller than n. Methods assuming this hypothesis are therefore often called manifold learning algorithms. Examples include PCA and nonlinear PCA [19], Isomap [32], and UMAP [22] . However, the manifold hypothesis does not always hold, especially when the space underlying the data contains singularities. In fact, singularities are ubiquitous in mathematics and appear often in physicals models [12, 28], hence there is a need to go beyond the manifold hypothesis.  \nTo transition from the smooth to the singular setting we replace the language of differential geometry and smooth manifolds with algebraic geometry and algebraic varieties. At a basic level, algebraic geometry studies the geometry of the zeros of systems of polynomials. Those zero sets are called algebraic varieties. At the heart of algebraic geometry lies the duality between geometry and algebra which allows us to jump back and forth between geometric spaces and computable algebraic procedures. Furthermore, algebraic geometry offers a natural setting for studying and working with singularities. As we shall see, this variety hypothesis provides a great amount of flexibility and it lends itself well to computations.  \nTo examine data in the singular setting we introduce the algebraic machine learning pipeline depicted in Figure 1.1 . This pipeline combines ideas from algebraic geometry and machine learning and it is the main subject of this paper.  \nUniversity of Colorado UCB 395, Department of Mathematics, Boulder CO 80309, USA  \nE-mail addresses: [ezzeddine.elsai@colorado.edu](ezzeddine.elsai@colorado.edu) , [parker.gara@colorado.edu](parker.gara@colorado.edu) , [markus.pflaum@colorado.edu](markus.pflaum@colorado.edu) .  \n2 ALGEBRAIC MACHINE LEARNING WITH AN APPLICATION TO CHEMISTRY  \nAlgebraic Computations  \nData  \nGeometric Insights  \nNumerical Computations  \nFigure 1.1  \nIn Section 2 we ","cbCailPt1KIsGLTp","https://ap.wps.com/l/cbCailPt1KIsGLTp","pdf",3338098,1,18,"English","en",105,"# Introduction\n# Algebraic computations\n## Learning problem and MAP formulation\n## Gröbner basis and geometric invariants\n# Numerical computations\n## Singularity heuristic and testing on synthetic and chemical data\n# Related work","[{\"question\":\"Why do manifold learning methods fail in some scientific applications?\",\"answer\":\"Manifold learning relies on the manifold hypothesis that data lie on a smooth submanifold. The assumption breaks down when the underlying space contains singularities, which are common in mathematical and physical models.\"},{\"question\":\"What is the core idea of the proposed algebraic machine learning pipeline?\",\"answer\":\"Instead of differential geometry and smooth manifolds, the pipeline works in algebraic geometry using algebraic varieties, aiming to recover the underlying variety from sampled data.\"},{\"question\":\"How is the variety-learning problem solved computationally?\",\"answer\":\"The learning problem is cast as a Maximum A Posteriori (MAP) optimization problem that is solved in terms of an eigenvalue computation, enabling recovery of the underlying variety.\"}]","ALGEBRAIC MACHINE LEARNING WITH AN APPLICATION TO CHEMISTRY | PDF",1785892995,45,{"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},"algebraic-machine-learning-with-an-application-to-chemistry","",{"@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/algebraic-machine-learning-with-an-application-to-chemistry/124562/",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","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why do manifold learning methods fail in some scientific applications?","Question",{"text":75,"@type":76},"Manifold learning relies on the manifold hypothesis that data lie on a smooth submanifold. The assumption breaks down when the underlying space contains singularities, which are common in mathematical and physical models.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the core idea of the proposed algebraic machine learning pipeline?",{"text":80,"@type":76},"Instead of differential geometry and smooth manifolds, the pipeline works in algebraic geometry using algebraic varieties, aiming to recover the underlying variety from sampled data.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the variety-learning problem solved computationally?",{"text":84,"@type":76},"The learning problem is cast as a Maximum A Posteriori (MAP) optimization problem that is solved in terms of an eigenvalue computation, enabling recovery of the underlying variety.","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"]