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Big Data Mining and Analytics  2020, Vol. 3 Issue (4): 235-258    DOI: 10.26599/BDMA.2020.9020011
    
Survey on Lie Group Machine Learning
Mei Lu(),Fanzhang Li*()
School of Software Engineering, Jinling Institute of Technology, Nanjing 211169, China and is also with the School of Computer Science and Technology, Jiangsu Normal University, Xuzhou 221000, China
School of Computer Science and Technology, Soochow University, Suzhou 215006, China
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Abstract  

Lie group machine learning is recognized as the theoretical basis of brain intelligence, brain learning, higher machine learning, and higher artificial intelligence. Sample sets of Lie group matrices are widely available in practical applications. Lie group learning is a vibrant field of increasing importance and extraordinary potential and thus needs to be developed further. This study aims to provide a comprehensive survey on recent advances in Lie group machine learning. We introduce Lie group machine learning techniques in three major categories: supervised Lie group machine learning, semisupervised Lie group machine learning, and unsupervised Lie group machine learning. In addition, we introduce the special application of Lie group machine learning in image processing. This work covers the following techniques: Lie group machine learning model, Lie group subspace orbit generation learning, symplectic group learning, quantum group learning, Lie group fiber bundle learning, Lie group cover learning, Lie group deep structure learning, Lie group semisupervised learning, Lie group kernel learning, tensor learning, frame bundle connection learning, spectral estimation learning, Finsler geometric learning, homology boundary learning, category representation learning, and neuromorphic synergy learning. Overall, this survey aims to provide an insightful overview of state-of-the-art development in the field of Lie group machine learning. It will enable researchers to comprehensively understand the state of the field, identify the most appropriate tools for particular applications, and identify directions for future research.



Key wordsLie group machine learning      Lie group subspace orbit generation learning      quantum group learning      symplectic group learning      Lie group fiber bundle learning     
Received: 14 May 2020      Published: 07 December 2020
Fund:  National Key Research and Development Program(Nos. 2018YFA0701700 and 2018YFA0701701);Scientific Research Foundation for Advanced Talents(No. jit-b-202045)
Corresponding Authors: Fanzhang Li     E-mail: louisazhaoxiaolu@sina.com;lfzh@suda.edu.cn
About author: Mei Lu received the PhD degree in computer science and technology from Soochow University in 2016. Now she is an associate professor and master supervisor at Jiangsu Normal University. Her main research interests include machine learning, data mining, big data science and engineering, etc.|Fanzhang Li received the MS degree in computer science and technology from University of Science and Technology of China in 1995. Now he is a professor and PhD supervisor at Soochow University. His main research interests include Lie group machine learning, big data science and engineering, dynamic fuzzy logic, etc.
Cite this article:

Mei Lu,Fanzhang Li. Survey on Lie Group Machine Learning. Big Data Mining and Analytics, 2020, 3(4): 235-258.

URL:

http://bigdata.tsinghuajournals.com/10.26599/BDMA.2020.9020011     OR     http://bigdata.tsinghuajournals.com/Y2020/V3/I4/235

NameSymbolLie algebra
Euclidean spaceinin
General linear groupGL(n, F)gl(n, F)
Special linear groupSL(n, F)sl(n, F)
Orthogonal groupO(n, F)o(n, F)
Special orthogonal groupSO(n, F)so(n, F)
Symplectic groupSP(n,F)sp(n, F)
Unitary groupU?(n)u?(n)
Special Unitary groupSU(n)su(n)
(p,q) type Unitary groupU(p, q)u(p, q)
Special (p,q) type UnitarySU(p, q)su(p, q)
Table 1 Parameter of Lie groups and Lie algebras.
Fig. 1 Example of a special orthogonal group: mappings between a Lie group and a Lie algebra using the exponential and logarithmic maps.
Fig. 2 Left effect model of Lie group machine learning.
Fig. 3 Right effect model of Lie group machine learning.
Fig. 4 Algebraic model of Lie group machine learning.
Fig. 5 Geometric model of Lie group machine learning.
Fig. 6 Diagram of learning space, learning subspace, orbit, and target orbit.
, here G is the covering group of Lie group G.
">
Fig. 7 Transformation from simply connected Lie group G to its covering group G, here G is the covering group of Lie group G.
Fig. 8 Linear dimensionality reduction algorithm for the dimensionality reduction of data.
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