[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117027-en":3,"doc-seo-117027-105":29,"detail-sidebar-cat-0-en-105":90},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},117027,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Polytopes and Machine Learning","Machine learning methods are developed for studying lattice polytopes, connecting data-driven prediction with established mathematical invariants. Supervised learning is used to forecast key properties such as volume, dual volume, and reflexivity, achieving accuracies reported as up to 100%. The approach targets 2D lattice polygons and 3D lattice polytopes using Plücker coordinates as inputs, showing improved performance over the conventional vertex representation and enabling data features tailored to polytope geometry.","LIMS-2021-011  \nPolytopes and Machine Learning  \nJiakang Baoa;b Yang-Hui Hea;b;c;d Edward Hirsta;b Johannes Hofscheiere Alexander Kasprzyke Suvajit Majumdera  \na Department of Mathematics, City, University of London, EC1V 0HB, UK  \nb London Institute for Mathematical Sciences, Royal Institution, London W1S 4BS, UK c Merton College, University of Oxford, OX1 4JD, UK  \nd School of Physics, NanKai University, Tianjin, 300071, P. R. China  \ne School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK E-mail: [jiakang. bao@city. ac. uk](jiakang. bao@city. ac. uk), [hey@maths. ox. ac. uk](hey@maths. ox. ac. uk),  \n[edward. hirst@city. ac. uk](edward. hirst@city. ac. uk), [johannes. hofscheier@nottingham. ac. uk](johannes. hofscheier@nottingham. ac. uk) ,  \n[a. m. kasprzyk@nottingham. ac. uk](a. m. kasprzyk@nottingham. ac. uk) , [suvajit. majumder@city. ac. uk](suvajit. majumder@city. ac. uk)  \nAbstract: We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, re􀀍exivity, etc, with accuracies up to 100% . We focus on 2d polygons and 3d polytopes with Pl􀁿ucker coordinates as input, which out-perform the usual vertex representation.  \nContents  \n1 Introduction 1  \n2 Polytope Preliminaries 3  \n2.1 Pl􀁿ucker Coordinates ............................ 5  \n3 Polytope Datasets 7  \n3.1 Polygons in dimension 2 .......................... 7  \n3.2 Polyhedra in dimension 3 .......................... 9  \n4 Machine Learning 10  \n4.1 Methodology ................................ 10  \n4.2 Polygons ................................... 11  \n4.2.1 Predicting Polygon Properties ................... 12  \n4.2.2 Pl􀁿ucker Augmentation ....................... 13  \n4.2.3 Restricting Training ........................ 15  \n4.2.4 The Inverse Problem ........................ 16  \n4.3 Polyhedra .................................. 17  \n4.3.1 Predicting Polyhedra Volumes ................... 17  \n4.3.2 Predicting Dual Volumes ...................... 20  \n4.3.3 Re􀀍exivity of the Polyhedra .................... 22  \n4.3.4 From Polygons to Polyhedra .................... 24  \n4.4 The MDS Embedding ............................ 26  \n5 Summary & Outlook 28  \nReferences 30  \n1 Introduction  \nLattice polytopes { convex bodies with integral vertices { lie in the intersection of combinatorics, optimisation theory, number theory, geometry, and theoretical physics. In combinatorics, problems include 􀀌nding lattice points inside a lattice polytope (for example, [1 , 2]) . In number theory, lattice polytopes constitute the cornerstone of the  \ngeometry of numbers (for example, [3 , 4]) . In geometry, lattice polytopes play a key role in the study of toric varieties, whose geometry can be described combinatorially in terms of cones and fans (for example,[5{8]) . In string theory, Batyrev{Borisov mirror symmetry [9] { describing how Calabi{Yau manifolds can be naturally constructed from a re􀀍exive polytope { initiated a large-scale collaboration between fundamental physics and computational algebraic geometry, culminating in the classi􀀌cation of re􀀍exive polytopes in dimensions three and four [10{12] . Details of these, and many other, applications of lattice polytopes can be found in [13 , 14] .  \nThere has been considerable recent interest in applying techniques from data science and machine learning (ML) to the study of pure mathematical data. Whilst initially this arose in large part from the investigation of the string theory/algebraic geometry landscape [15{20], it is natural to ask whether this paradigm can be applied to di􀀋erent disciplines within pure mathematics [21{26] .  \nCombinatorial geometry lends itself to ML. The central objects { lattice polytopes { can be cast into a matrix of integers and fed to a neural network (NN) classi􀀌er or regressor. We will discuss di􀀋erent representations of this input data in this paper. Meanwhile, much of th","cbCaiiBUTOJM2iua","https://ap.wps.com/l/cbCaiiBUTOJM2iua","pdf",2973622,1,34,"English","en",105,"# Introduction\n# Polytope Preliminaries\n## Plücker Coordinates\n# Polytope Datasets\n## Polygons in dimension 2\n## Polyhedra in dimension 3\n# Machine Learning\n## Methodology\n## Polygons\n### Predicting Polygon Properties\n### Plücker Augmentation\n### Restricting Training\n### The Inverse Problem\n## Polyhedra\n### Predicting Polyhedra Volumes\n### Predicting Dual Volumes\n### Reflexivity of the Polyhedra\n### From Polygons to Polyhedra\n## The MDS Embedding\n# Summary & Outlook\n# References","[{\"question\":\"What is the main goal of the work on polytopes and machine learning?\",\"answer\":\"To introduce machine learning techniques for lattice polytopes and predict standard polytope properties from numerical inputs. The focus is on learning mathematical structure rather than replacing the underlying mathematics.\"},{\"question\":\"Which input representation is used for 2D polygons and 3D polytopes?\",\"answer\":\"Plücker coordinates are used as the input features for both polygons (2D) and polytopes (3D). The paper emphasizes that this representation can outperform the usual vertex representation.\"},{\"question\":\"What polytope properties are predicted using supervised learning?\",\"answer\":\"The study predicts properties including volume, dual volume, and reflexivity. It also examines other invariants such as the Gorenstein index and codimension.\"}]",1785673146,86,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"polytopes-and-machine-learning","",{"@graph":35,"@context":84},[36,53,67],{"@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/polytopes-and-machine-learning/117027/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is the main goal of the work on polytopes and machine learning?","Question",{"text":74,"@type":75},"To introduce machine learning techniques for lattice polytopes and predict standard polytope properties from numerical inputs. The focus is on learning mathematical structure rather than replacing the underlying mathematics.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Which input representation is used for 2D polygons and 3D polytopes?",{"text":79,"@type":75},"Plücker coordinates are used as the input features for both polygons (2D) and polytopes (3D). The paper emphasizes that this representation can outperform the usual vertex representation.",{"name":81,"@type":72,"acceptedAnswer":82},"What polytope properties are predicted using supervised learning?",{"text":83,"@type":75},"The study predicts properties including volume, dual volume, and reflexivity. 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