[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121578-en":3,"doc-seo-121578-105":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},121578,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Logifold - A Geometrical Foundation of Ensemble Machine Learning","The work introduces a local-to-global, measure-theoretic framework for analyzing datasets through a newly defined logifold structure. It models network components with restricted domains as local charts, providing a mathematical foundation for ensemble machine learning. Experiments show logifolds can detect fuzzy domains and improve prediction accuracy versus naive averaging, where domain-agnostic averaging may degrade performance. A theoretical example clarifies why restricting classifier domains is essential in ensembles.","Proc. of the International Conference on Electrical, Computer, Communications and Mechatronics Engineering (ICECCME 2024)  \n4-6 November 2024, Male, Maldives  \nLogifold: A Geometrical Foundation of Ensemble  \nMachine Learning  \narXiv :2407 . 16177v2 [ cs .LG] 19 Oct 2024  \nInkee Jung  \nDepartment of Mathematics and Statistics Boston University Boston, MA, USA[inkeej@bu.edu](inkeej@bu.edu)  \nSiu-Cheong Lau  \nDepartment of Mathematics and Statistics Boston University Boston, MA, USA [scllouis@bu.edu](scllouis@bu.edu)  \nAbstract—We present a local-to-global and measure-theoretical approach to understanding datasets. The core idea is to formulatea logifold structure and to interpret network models with restricted domains as local charts of datasets. In particular, this provides a mathematical foundation for ensemble machine learning. Our experiments demonstrate that logifolds can be implemented to identify fuzzy domains and improve accuracy compared to taking average of model outputs. Additionally, we provide a theoretical example of a logifold, highlighting the importance of restricting to domains of classi􀀂ers in an ensemble.  \nIndex Terms—Local to global principle, Neural Network, Ensemble Machine Learning, Fuzziness  \nI. INTRODUCTION  \nThe concept of a manifold has been broadly used in data science for interpolating data points (for instance, [1] gives an excellent overview of the topic) . Recently, the study of dataset using topological methods develops into an interesting research area, see for instance [2], [3], [4] . In most applications, manifolds are understood as higher dimensional analogs of surfaces in the Euclidean space R3.  \nOn the other hand, a crucial aspect of a manifold is the local-to-global perspective to study spaces, which is often overlooked in applications to data science. In [5] and this paper, we would like to formulate a local-to-global approach to study datasets.  \nManifolds can be expressed as zero loci of smooth functions, which are well approximated by polynomials. In contrast, datasets are like ‘point clouds’ and not locally Euclidean. Thus, we propose to model a dataset by a measure space. Moreover, inspired by the huge success of neural networks, we take the graphs of linear logical functions as local models.  \nLinear logical functions are de􀀂ned via graphs and linear inequalities, whose targets are 􀀂nite sets. We will restrict their domains to measurable subsets of Rn. Functions obtained by arti􀀂cial neural networks belong to this class. Linear logical functions are universal, in the sense that they can approximate any measurable functions with a 􀀂nite target set. Consequently, constructing an atlas from the graphs of logical functions leads to an analog of a topological manifold in this setting, which we call to be a logifold.  \nFuzziness is another important aspect in our formulation. Logical functions used in machine learning exhibit the characteristics of fuzzy logic, with values in the range [0 , 1] rather than {0 , 1} . Thus, the graph of a logical function exhibit fuzziness, leading to the notion of a fuzzy logifold.  \nLogifold provides a mathematical foundation for ensemble machine learning. Ensemble machine learning takes a weighted average of several models. It has shown impressive results in classi􀀂cation problems ([6], [7]) . Moreover, it is shown to reduce bias and variance for clearer decision boundaries ([8], [9], [10], [11]) .  \nOur logifold formulation emphasizes the importance of restricting to the domain of each model when we take average. Otherwise wrong predictions of a model outside its domain can seriously harm the average accuracy. We will make a theoretical example of a logifold to show the limitation of averaging over models.  \nIn practice, the certainty scores given by the softmax function provide some information about the domain of a model. This will be the main ingredient of implementing alogifold in practice. However, certainty score only contains partial informat","cbCaiqU9yaxAiis2","https://ap.wps.com/l/cbCaiqU9yaxAiis2","pdf",161691,1,6,"English","en",105,"# Introduction\n## Local-to-global perspective for datasets\n## Manifold vs. dataset modeling via measure spaces\n## Linear logical functions and atlas construction\n## Fuzzy logifolds and ensemble foundations\n## Domain restriction and practical implementation\n# Linear Logical Function","[{\"question\":\"What is the main concept of logifold in this paper?\",\"answer\":\"The paper formulates datasets using a logifold structure that interprets network models with restricted domains as local charts, enabling a local-to-global understanding of data.\"},{\"question\":\"How does logifold relate to ensemble machine learning?\",\"answer\":\"Logifold provides a mathematical foundation for ensembles by emphasizing that averaging should be performed only within each model’s valid domain, otherwise inaccurate out-of-domain predictions can harm accuracy.\"},{\"question\":\"What does the paper mean by fuzzy logifolds?\",\"answer\":\"Fuzzy logifolds arise because logical functions used in machine learning produce outputs in [0,1], so the function graphs reflect fuzziness and yield fuzzy domain behavior.\"}]","Logifold - 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