School of Software Engineering, Jinling Institute of Technology, Nanjing211169, China and is also with the School of Computer Science and Technology, Jiangsu Normal University, Xuzhou221000, China School of Computer Science and Technology, Soochow University, Suzhou215006, China

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.

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.

Fig. 7Transformation from simply connected Lie group G to its covering group G${}^{\mathbf{\prime}}$, here $G{}^{\mathbf{\prime}}$ is the covering group of Lie groupG.

Fig. 8Linear dimensionality reduction algorithm for the dimensionality reduction of data.

[91]

Huynh-The T., Hua C. H., Ngo T. T., and Kim D. S., Image representation of pose-transition feature for 3D skeleton-based action recognition, Inf. Sci., vol. 513, pp. 112-126, 2020.

[92]

Vemulapalli R., Arrate F., and Chellappa R., Human action recognition by representing 3D skeletons as points in a lie group, in Proc. 2014 IEEE Conf. on Computer Vision and Pattern Recognition, Columbus, OH, USA, 2014, p. 14 632 342.

[93]

Liu Z., Zhang C. Y., and Tian Y. L., 3D-based deep convolutional neural network for action recognition with depth sequences, Image Vision Comput., vol. 55, pp. 93-100, 2016.

[94]

Cai X. Y., Zhou W. G., and Li H. Q., Attribute mining for scalable 3D human action recognition, in Proc. 23rd ACM Int. Conf. on Multimedia, Brisbane, Australia, 2015, pp. 1075-1078.

[95]

Zhang H. L., Zhong P., He J. L., and Xia C. X., Combining depth-skeleton feature with sparse coding for action recognition, Neurocomputing, vol. 230, pp. 417-426, 2017.

[96]

Nú?ez J. C., Cabido R., Pantrigo J. J., Montemayor A. S., and Vélez J. F., Convolutional neural networks and long short-term memory for skeleton-based human activity and hand gesture recognition, Pattern Recognit., vol. 76, pp. 80-94, 2018.

[97]

Demisse G. G., Aouada D., and Ottersten B., Deformation based curved shape representation, IEEE Trans. Pattern Anal. Mach. Intell., vol. 40, no. 6, pp. 1338-1351, 2018.

[98]

Von Tycowicz C., Ambellan F., Mukhopadhyay A., and Zachow S., An efficient riemannian statistical shape model using differential coordinates: With application to the classification of data from the osteoarthritis initiative, Med. Image Anal., vol. 43, pp. 1-9, 2018.

[99]

Hayat M., Bennamoun M., and El-Sallam A. A., An RGB-D based image set classification for robust face recognition from Kinect data, Neurocomputing, vol. 171, pp. 889-900, 2016.

[100]

Yin M., Wu Z. Z., Shi D. M., Gao J. B., and Xie S. L., Locally adaptive sparse representation on riemannian manifolds for robust classification, Neurocomputing, vol. 310, pp. 69-76, 2018.

[101]

Wang T., Qiao M. N., Chen Y., Chen J., Zhu A. C., and Snoussi H., Video feature descriptor combining motion and appearance cues with length-invariant characteristics, Optik, vol. 157, pp. 1143-1154, 2018.

[102]

Dong J. M., Peng Y. X., Ying S. H., and Hu Z. Y., Lietricp: An improvement of trimmed iterative closest point algorithm, Neurocomputing, vol. 140, pp. 67-76, 2014.

[1]

Li F. Z., He S. P., and Qian X. P., Survey on lie group machine learning, (in Chinese), Chin. J. Comput., vol. 33, no. 7, pp. 1115-1126, 2010

[2]

Li F. Z., Qian X. P., Xie L., and He S. P., Machine Learning Theory and Its Applications, (in Chinese). Hefei, China: China University of Science and Technology Press, 2009.

[103]

Ying S. H., Wang Y. W., Wen Z. J., and Lin Y. P., Nonlinear 2D shape registration via thin-plate spline and lie group representation, Neurocomputing, vol. 195, pp. 129-136, 2016.

[3]

Li F. Z., Zhang L., Yang J. W., Qian X. P., Wang B. J., and He S. P., Lie Group Machine Learning, (in Chinese). Hefei, China: China University of Science and Technology Press, 2013.

[4]

Li F. Z., Zhang L., and Zhang Z., Lie Group Machine Learning, , 2019.

[5]

Haill T. A., An application of the lie group theory of continuous point transformations to the Vlasov-Maxwell equations (Plasma Physics), PhD dissertation, University of Illinois at Urbana-Champaign, Champaign, IL, USA, 1985.

[6]

Baker A., Matrix Groups: An Introduction to Lie Group Theory. Berlin, Germany: Springer, 2002.

[7]

Fulton W. and Harris J., Representation theory: A first course, , 1991.

[8]

Hunacek M., Lie groups: An introduction through linear groups, , 2008.

[9]

Yarlagadda P., Ozcanli O., and Mundy J., Lie group distance based generic 3-d vehicle classification, in Proc. 2008 19th Int. Conf. on Pattern Recognition, Tampa, FL, USA, 2008.

[10]

Lui Y. M., Advances in matrix manifolds for computer vision, Image Vision Comput., vol. 30, nos. 6&7, pp. 380-388, 2012.

[11]

Xu H. and Li F. Z., The design of Su(n) classifier of lie group machine learning (LML), J. Comput. Informat. Syst., vol. 1, no. 4, pp. 835-841, 2005.

[12]

Xu H. and Li F. Z., Algorithms of dynkin diagrams in lie group machine learning, J. Commun. Comput., vol. 4, no. 3, pp. 13-17, 2007.

[13]

Xu H. and Li F. Z., Geometry algorisms of dynkin diagrams in lie group machine learning, J. Nanchang Inst. Technol., vol. 25, no. 2, pp. 74-78, 2006.

[14]

Xu H. and Li F. Z., Lie group machine learning’s axiom hypothesizes, in Proc. 2006 IEEE Int. Conf. on Granular Computing, Atlanta, GA, USA, 2006, pp. 401-404.

[15]

Chen F., Research and application on orbits generated algorithm of learning subspace in Lie-Group Machine Learning (LML), (in Chinese), master dissertation, Soochow University, Suzhou, China, 2007.

[16]

Chen F. and Li F. Z., Orbits generated theory of learning subspace and its algorithm in Lie-Group machine learning (LML), J. Suzhou Univ. (Nat. Sci. Ed.), vol. 23, no. 1, pp. 61-66, 2007.

[17]

Lv C. L., Wu Z. K., Zhang D., Wang X. C., and Zhou M. Q., 3D nose shape net for human gender and ethnicity classification, Pattern Recognit. Lett., vol. 126, pp. 51-57, 2019.

[18]

Boutellaa E., Kerdjidj O., and Ghanem K., Covariance matrix based fall detection from multiple wearable sensors, J. Biomed. Informat., vol. 94, p. 103 189, 2019.

[19]

Heider Y., Wang K., and Sun W. C., SO(3)-invariance of informed-graph-based deep neural network for anisotropic elastoplastic materials, Comput. Methods Appl. Mech. Eng., vol. 363, p. 112 875, 2020.

[20]

Lebanon G., Metric learning for text documents, IEEE Trans. Pattern Anal. Mach. Intell., vol. 28, no. 4, pp. 497-508, 2006.

[21]

Chen F. and Li F. Z., Orbits generated lattice algorithm of learning subspace in Lie-group Machine Learning (LML), (in Chinese), Comput. Eng. Appl., vol. 43, no. 15, pp. 184-187, 2007.

[22]

Nasios N. and Bors A. G., Kernel-based classification using quantum mechanics, Pattern Recognit., vol. 40, no. 3, pp. 875-889, 2007.

[23]

He S. P. and Li F. Z., A molecular docking drug design algorithm based on quantum group, (in Chinese), J. Nanjing Univ. (Nat. Sci.), vol. 44, no. 5, pp. 512-519, 2008.

[24]

Wu Y. M., Hu J. Y., and Yin X. G., Symplectic integrators of the equations of multibody system dynamics on manifolds, (in Chinese), Adv. Mech., vol. 32, no. 2, pp. 189-195, 2002.

[25]

Feng K. and Qin M. Z., Symplectic Geometric Algorithms for Hamiltonian Systems. Berlin, Germany: Springer, 2010.

[26]

Xu Z. X., Zhou D. Y., and Deng Z. C,, Numerical method based on hamilton system and symplectic algorithm to differential games, Appl. Math. Mech., vol. 27, no. 3, pp. 341-346, 2006.

[27]

Fu H. X. and Li F. Z., Research of the symplectic group classifier based on lie group machine learning, in Proc. 4th Int. Conf. on Fuzzy Systems and Knowledge Discovery (FSKD 2007), Haikou, China, 2007, pp. 649-655.

[28]

Fu H. X., Research on symplectic group classifier in Lie group machine learning, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2008.

[29]

Guan W. W. and Li F. Z., Drug molecular design using lie group machine learning (LML), in Proc. Int. Conf. on Advanced Intelligence, Beijing, China, 2008, pp. 411-414.

[30]

Nechaeva O., The neural network approach to automatic construction of adaptive meshes on multiply-connected domains, in Proc. Int. Joint Conf. on Neural Network, Orlando, FL, USA, 2007, pp. 1912-1917.

[31]

Guan W. W., Research of covering algorithm in Lie Group machine learning, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2009.

[32]

Lui L. M., Zeng W., Yau S. T., and Gu X. F., Shape analysis of planar objects with arbitrary topologies using conformal geometry, in Proc. 11th European Conf. on Computer Vision, Crete, Greece, 2010, pp. 672-686.

[33]

Chen Y., Li F. Z., and Zou P., Multiply connected lie group covering learning algorithm for image classifi-cation, (in Chinese), J. Front. Comput. Sci. Technol., vol. 8, no. 9, pp. 1101-1112, 2014.

[34]

Yan C. and Li F. Z., Path optimization algorithms for covering learning, (in Chinese), J. Software, vol. 26, no. 11, pp. 2781-2794, 2015.

[35]

Wu L. H. and Li F. Z., Multiply Lie Group kernel covering learning algorithm for image classification, (in Chinese), J. Front. Comput. Sci. Technol., vol. 10, no. 12, pp. 1737-1743, 2016.

[36]

Bengio Y. S., Learning deep architectures for AI, Found. Trends Mach. Learn., vol. 2, no. 1, pp. 1-127, 2009.

[37]

Hinton G. E., Osindero S., and Teh Y. W., A fast learning algorithm for deep belief nets, Neural Comput., vol. 18, no. 7, pp. 1527-1554, 2006.

[38]

He W. H. and Li F. Z., Research on Lie Group deep structure learning algorithm, (in Chinese), J. Front. Comput. Sci. Technol., vol. 4, no. 7, pp. 646-653, 2010.

[39]

Hong Z. Q., Algebraic feature extraction of image for recognition, Pattern Recognit., vol. 24, no. 3, pp. 211-219, 1991.

[40]

Tan Y., Tan T. N., Wang Y. H., and Fang Y. C., Do singular values contain adequate information for face recognition? Pattern Recognit., vol. 36, no. 3, pp. 649-655, 2003.

[41]

Yang M. D., Li F. Z., Zhang L., and Zhang Z., Lie group impression for deep learning, Informat. Process. Lett., vol. 136, pp. 12-16, 2018.

[42]

Gao C., Li F. Z., and Shen C., Research on Lie Group kernel learning algorithm, (in Chinese), J. Front. Comput. Sci. Technol., vol. 6, no. 11, pp. 1026-1038, 2012.

[43]

Govindu V. M., Lie-algebraic averaging for globally consistent motion estimation, in Proc. 2004 IEEE Computer Society Conf. on Computer Vision and Pattern Recognition, Washington, DC, USA, 2004, p. 8 161 455.

[44]

Subbarao R. and Meer P., Nonlinear mean shift for clustering over analytic manifolds, in Proc. IEEE Computer Society Conf. on Computer Vision and Pattern Recognition, New York, NY, USA, 2006.

[45]

Tuzel O., Subbarao R., and Meer P., Simultaneous multiple 3D motion estimation via mode finding on lie groups, in Proc. 10th IEEE Int. Conf. on Computer Vision, Beijing, China, 2005, pp. 18-25.

[46]

Moakher M., Means and averaging in the group of rotations, SIAM J. Matrix Anal. Appl., vol. 24, no. 1, pp. 1-16, 2002.

[47]

Gao C. and Li F. Z., Lie group means learning algorithm, (in Chinese), Pattern Recognit. Artif. Intell., vol. 25, no. 6, pp. 900-908, 2012.

[48]

Gao C., Research on Lie Group mean learning algorithm and its application, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2012.

[49]

Sch?lkopf B., Smola A., and Müller K. R., Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation, vol. 10, no. 5, pp, 1299-1319, 1998.

[50]

Aizerman M. A., Braverman E. M., and Rozonoer L. I., Theoretical foundations of the potential function method in pattern recognition learning, Automat. Remote Control, vol. 25, no. 6, pp. 821-837, 1964.

[51]

Cheng J., Liu Q. S., and Lu H. Q., Texture classification using kernel independent component analysis, in Proc. 17th Int. Conf. on Pattern Recognition, Cambridge, UK, 2004, p. 8 213 183.

[52]

Mika S., Ratsch G., Weston J., Scholkopf B., and Mullers K. R., Fisher discriminant analysis with kernels, in Neural Networks for Signal Processing IX: Proc. 1999 IEEE Signal Processing Society Workshop, Madison, WI, USA, 1999, p. 6 497 095.

[53]

Xu C. Y., Lu C. Y., Gao J. B., Wang T. J., and Yan S. C., Facial analysis with a lie group kernel, IEEE Trans. Circuits Syst. Video Technol., vol. 25, no. 7, pp. 1140-1150, 2015.

[54]

Sheehan B. N. and Saad Y., Higher order orthogonal iteration of tensors (HOOI) and its relation to PCA and GLRAM, in Proc. SIAM Int. Conf. on Data Mining, Minneapolis, MN, USA, 2007.

[55]

Yang J., Zhang D., Frangi A. F., and Yang J. Y., Two-dimensional PCA: A new approach to appearance-based face representation and recognition, IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 1, pp. 131-137, 2004.

[56]

Ye J. P., Generalized low rank approximations of matrices, Mach. Learn., vol. 61, no. 1, pp. 167-191, 2005.

[57]

Lu M. and Li F. Z., Neighborhood-embedded tensor learning, (in Chinese), J. Front. Comput. Sci. Technol., vol. 11, no. 7, pp. 1102-1113, 2017.

[58]

Sun J. T., Zeng H. J., Liu H., Lu Y. C., and Lu Y., CubeSVD: A novel approach to personalized web search, in Proc. 14th Int. Conf. on World Wide Web, Chiba, Japan, 2005.

[59]

He X. F., Cai D., and Niyogi P., Tensor subspace analysis, in Proc. 18th Int. Conf. on Neural Information Processing Systems, British Columbia, Canada, 2005, pp. 499-506.

[60]

Li X. L., Research and application on a data reduction method based on tensor field, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2009.

[61]

Lu C. T., Research on tensor learning algorithm and its application on disease prediction, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2017.

[62]

Tang J. H., Shu X. B., Li Z. C., Jiang Y. G., and Tian Q., Social anchor-unit graph regularized tensor completion for large-scale image retagging, IEEE Trans. Pattern Anal. Mach. Intell., vol. 41, no. 8, pp. 2027-2034, 2019.

[63]

Li J., Li F. Z., and Zhang L., Image segmentation algorithm based on affine connection, (in Chinese), J. Front. Comput. Sci. Technol., vol. 6, no. 3, pp. 267-274, 2012.

[64]

Xu Q., Li F. Z., and Zou P., The k-means algorithm based on finsler geometry, (in Chinese), J. Univ. Sci. Technol. China, vol. 7, pp. 570-575, 2014.

[65]

Chen M., He S. P., and Li F. Z., A lie group machine learning algorithm for linear classification, (in Chinese), Microelectron. Comput., vol. 26, no. 10, pp. 170-173, 2009.

[66]

Xian M. and Li F. Z., Learning algorithm based on homology boundary, (in Chinese), Comput. Eng. Appl., vol. 44, no. 21, pp. 192-194&204, 2008.

[67]

Aiolli F. and Sperduti A., A re-weighting strategy for improving margins, Artif. Intell., vol. 137, nos. 1&2, pp. 197-216, 2002.

[68]

Canny J., A computational approach to edge detection, IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-8, no. 6, pp. 679-698, 1986.

[69]

Zhao M. M. and Li F. Z., Neighborhood homology learning algorithm, (in Chinese), CAAI Trans. Intell. Syst., vol. 9, no. 3, pp. 336-342, 2014.

[70]

Perperidis A., Dhaliwal K., McLaughlin S., and Vercauteren T., Image computing for fibre-bundle endomicroscopy: A review, Med. Image Anal., vol. 62, p. 101 620, 2020.

[71]

Sochen N., Kimmel R., and Malladi R., A general framework for low level vision, IEEE Trans. Image Process., vol. 7, no. 3, pp. 310-318, 1998.

[72]

Pearson D. W., Approximating vertical vector fields for feedforward neural networks, Appl. Math. Lett., vol. 9, no. 2, pp. 61-64, 1996.

[73]

Chao J. H. and Kim J., A fibre bundle model of surfaces and its generalization, in Proc. 17th Int. Conf. on Pattern Recognition, Cambridge, UK, 2004, p. 8 221 516.

[74]

Zhou L. L. and Li F. Z., Research on mapping mechanism of learning expression, in Proc. 5th Int. Conf. on Rough Set and Knowledge Technology, Beijing, China, 2010, pp. 298-303.

[75]

Gao T. R., The diffusion geometry of fibre bundles: Horizontal diffusion maps, Appl. Comput. Harmonic Anal., .
doi: 10.1016/j.acha.2019.08.001

[76]

Asperti A. and Longo G., Categories Types and Structures: An Introduction to Category Theory for the Working Computer Scientist. Cambridge, MA, USA: MIT Press, 1991.

[77]

Muhiuddin G., Basic Concepts of Category Theory and Its Applications. LAP LAMBERT Academic Publishing, 2018.

[78]

Xu X. X., Li F. Z., Zhang L., and Zhang Z., The category representation of machine learning algorithm, (in Chinese), J. Comput. Res. Dev., vol. 54, no. 11, pp. 2567-2575, 2017.

[79]

Lu M., Zhang L., Zhao X. J., and Li F. Z., Constrained neighborhood preserving concept factorization for data representation, Know. Based Syst., vol. 102, pp. 127-139, 2016.

[80]

Xu H. X., A semi-supervised learning algorithm based on Lie Group and application, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2009.

[81]

He W., Research on the covering algorithm of machine learning and its application, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2011.

[82]

Dong M. X., Spectral estimation of image features manifold learning method, (in Chinese), master dissertation, Soochow University, Suzhou, China, 2012.

[83]

Yang M. D., Li F. Z., and Zhang L., Advances in the study of lie group machine learning in recent ten years, (in Chinese), Chin. J. Comput., vol. 38, no. 7, pp. 1337-1356, 2015.

[84]

Huang Y. J. and Li F. Z., Isospectral manifold learning algorithm, (in Chinese), J. Softw., vol. 24, no. 11, pp. 2656-2666, 2013.

[85]

Huang Y. J. and Li F. Z., Fast learning algorithm of spectral manifold, (in Chinese), J. Front. Comput. Sci. Technol., vol. 8, no. 6, pp. 735-742, 2014.

[86]

Ren Y., Wang Y. N., and Zhu J., Spectral learning for supervised topic models, IEEE Trans. Pattern Anal. Mach. Intell., vol. 40, no. 3, pp. 726-739, 2018.

[87]

Tao D. C., Li X. L., Wu X. D., Hu W. M., and Maybank S. J., Supervised tensor learning, Knowl. Inf. Syst., vol. 13, no. 1, pp. 1-42, 2007.

[88]

Fei W., Liu Y. N., and Zhuang Y. T., Tensor-based transductive learning for multimodality video semantic concept detection, IEEE Trans. Multimed., vol. 11, no. 5, pp. 868-878, 2009.

[89]

Liu X. L., Guo T. J., He L. F., and Yang X. W., A low-rank approximation-based transductive support tensor machine for semisupervised classification, IEEE Trans. Image Process., vol. 24, no. 6, pp. 1825-1838, 2015.

[90]

Hu W. M., Gao J., Xing J. L., Zhang C., and Maybank S., Semi-supervised tensor-based graph embedding learning and its application to visual discriminant tracking, IEEE Trans. Pattern Anal. Mach. Intell., vol. 39, no. 1, pp. 172-188, 2017.