[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118007-en":3,"doc-seo-118007-105":30,"detail-sidebar-cat-0-en-105":92},{"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},118007,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Machine Learning - Clifford invariants of ADE Coxeter elements","Recent interest in Clifford geometric invariants motivates studying such invariants for Coxeter transformations arising in root systems, reflection groups, Lie groups, and Lie algebras. Exhaustive high-performance computations determine all Coxeter transformations for A8, D8, and E8 for a chosen basis of simple roots and yield their invariants. The resulting dataset is then suitable for mining with data-science methods, focusing here on neural-network classification and principal component analysis.","arXiv :2310 .00041v2 [ cs .LG] 26 May 2024  \nMachine Learning Clifford invariants of ADE Coxeter elements  \nSiqi Chen, Pierre-Philippe Dechant, Yang-Hui He, Elli Heyes, Edward Hirst and Dmitrii Riabchenko  \nTo Jim Humphreys, Peter Neumann, and John McKay  \nAbstract. There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A8 , D8 and E8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis.  \nSince the output – the invariants – is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.  \nMathematics Subject Classification (2010). Primary 52B15; Secondary 52B11, 15A66, 20F55, 17B22, 20G41 .  \nKeywords. Exceptional symmetries, invariants, Cayley-Hamilton theorem, Clifford algebras, Coxeter groups, root systems, Platonic solids.  \nReport Number. QMUL-PH-23-15 .  \n1. Introduction  \nGreat interest in Clifford geometric invariants of linear transformations, originally proposed in [45], was sparked in recent work from a practical [52,53] and theoretical [1,33,52,63] point of view. Orthogonal transformations, such as rotations, and their invariants are important in engineering, e.g. moving cameras, robots etc. The types of transformations we are looking at in this work are also rotations, particularly interesting because of their symmetry structures. Linear transformation invariants are traditionally exemplified by the determinant and trace, which appear in the highest and lowest coefficients of the characteristic polynomial. Typically such linear transformations are described by matrices; however, in Clifford algebras one has the alternative to implement orthogonal transformations via versors. In Clifford algebras, algebraic objects have a clearer geometric interpretation than in the standard matrix approach. There is a systematic way of calculating multivector invariants of linear transformations via what are called ‘simplicial derivatives’, which we will introduce further in  \n2 Chen, Dechant, He, Heyes, Hirst and Riabchenko  \nthe next section. These Clifford geometric invariants are then systematically related to geometric invariant spaces of the linear transformation and the coefficients in the characteristic polynomial and Cayley-Hamilton theorem 1 . This serves as motivation to study this type of new geometric invariant of linear transformations.  \nFrom the perspective of some of our other work on root systems and reflection groups [28,30,31] we are particularly interested in a certain type of linear transformations that occurs in this root system context: the ‘Coxeter elements’ or ‘Coxeter transformations’. These are a particular type of orthogonal transformation in reflection/Coxeter groups [47] . They are the group elements of the highest order (called the ‘Coxeter number’, h) and they are all conjugate to each other. High-dimensional root systems are n","cbCairf7MY3Medzm","https://ap.wps.com/l/cbCairf7MY3Medzm","pdf",1993524,1,34,"English","en",105,"# Introduction\n## Clifford geometric invariants and simplicial derivatives\n## Coxeter elements in root systems\n## Experimental mathematics via computational algebra and HPC","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It investigates Clifford geometric invariants of Coxeter transformations associated with root systems, reflection groups, and related Lie-theoretic structures.\"},{\"question\":\"How are the Coxeter invariants obtained in the study?\",\"answer\":\"The authors perform exhaustive high-performance computing calculations of all Coxeter transformations for A8, D8, and E8, then compute the corresponding invariants from the chosen simple-root basis.\"},{\"question\":\"Which machine-learning techniques are used?\",\"answer\":\"The study applies neural network classification and principal component analysis to learn from the computed invariant datasets.\"}]","Machine Learning - 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