张量树学习算法

南京大学学报(自然科学版) ›› 2015, Vol. 51 ›› Issue (2): 390–404.

张量树学习算法

路梅1,2李凡长1

出版日期:2015-03-06 发布日期:2015-03-06
作者简介:(苏州大学计算机学院苏州215006;2. 江苏师范大学计算机学院徐州221116)
基金资助:
国家自然科学基金(61033013,61272297,61402207 )

Tensor ree learning method

Lu Mei1,2, Li Fanzhang1

Online:2015-03-06 Published:2015-03-06
About author:(1. College of Computer Science and Technology, Soochow University, Suzhou, 215006, China; 2. College of Computer Science and Technology, Jiangsu Normal University, Xuzhou, 221116,China)

摘要/Abstract

摘要： 基于张量几何理论及人类视觉认知的一、二、三维认知模式，本文提出了张量树学习算法(Tensor Tree Learning, TTL)。其内容包括：张量树学习的基本概念、张量树学习算法、基于张量树的Tucker分解和CP分解的学习算法等；同时也给出了阶张量树树高的最小高度为；最后在数据库Coil100，Coil20和本实验室创建的数据库上进行了验证，结果表明张量树学习算法是有效、合理的。

Abstract: Based on the research of the tensor geometry theory and 1-D, 2-D and 3-D cognitive model of human visual perception, the Tensor Tree Learning (TTL) method is proposed in this paper. The main purpose of this paper are fourfold: the basic concept of tensor tree learning, TTL algorithm, Tucker decomposition and CP decomposition based on tensor tree algorithm. Furthermore, the minimum depth of the n-order tensor tree is also proven to be . The proposed method is evaluated on COIL100, COIL20,ORL and our lab’s databases to show its effectiveness and the results have proven the validity and rationality of our proposed TTL method

路梅1,2李凡长1. 张量树学习算法
[J]. 南京大学学报(自然科学版), 2015, 51(2): 390–404.

Lu Mei1,2, Li Fanzhang1. Tensor ree learning method[J]. Journal of Nanjing University(Natural Sciences), 2015, 51(2): 390–404.

参考文献

[1] 李凡长, 张莉, 杨季文, 等. 李群机器学习. 合肥：中国科学技术大学出版社, 2013.
[2] Hutchinson Brian, Deng Li, Yu Dong. Tensor deep stacking networks. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 2012, 35(8): 1944~1957.
[3] Tao D C, Li X L, Hu W M, et al. Supervised tensor learning. Knowl. Inf. Syst, 2007, 13:1~42.
[4] Guo W W, Kotsia I, Patras I. Tensor learning for regression. Image Processing, IEEE Transactions on, 2012, 21(2): 816~827.
[5] Bromuri S. A Tensor Factorization Approach to Generalization in Multi-agent Reinforcement Learning. in: Proceedings of the 2012 IEEE/WIC/ACM International Joint Conferences on Web Intelligence and Intelligent Agent Technology, 2012: 274~281.
[6] Duan G F, Wang H C, Liu Z Y, et al. K-CPD: Learning of overcomplete dictionaries for tensor sparse coding. in: Pattern Recognition (ICPR), 2012 21st International Conference on, 2012: 493~496.
[7] Shi Z Q, Han J Q, Zheng T R, et al. Audio Segment Classification Using Online Learning Based Tensor Representation Feature Discrimination. 2013, 21(1): 186~196.
[8] Zubair S, Wang W W. Tensor Dictionary Learning With Sparse Tucker Decomposition. in: Digital Signal Processing (DSP), 3013 18th International Conference on, 2013:1~6.
[9] Kung S Y. From green computing to big-data learning: A kernel learning perspective. in: Application-Specific Systems, Architectures and Processors (ASAP), 2013 IEEE 24th International Conference on, 2013: 1~1.
[10] Hao Z F, He L F, Chen B Q, et al. A Linear Support Higher-Order Tensor Machine for Classification. IEEE Transactions on Image Processing, 2013, 22: 2911~2920.
[11] Tucker L R. Implications of factor analysis of three-way matrices for measurement of change. Problems in measuring change, 1963: 122~137.
[12] Kroonenberg P M, De L J. Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 1980, 45(1): 69~97.
[13] Tucker L R. Some mathematical notes on three-mode factor analysis. Psychometrika, 1966, 31(3): 279~311.
[14] Krooneberg P M. Three-mode principal component analysis: Theory and applications. DSWO press: Leiden, 1983, 2.
[15] De L L, De M B, Vandewalle J. A multilinear singular value decomposition. SIAM journal on Matrix Analysis and Applications, 2000, 21(4): 1253~1278.
[16] De L L, De M B, Vandewalle J. On the best rank-1 and rank-(R1, R2,..., Rn) approximation of higher-order tensors. SIAM journal on Matrix Analysis and Applications, 2000, 21(4): 1324~1342.
[17] Cichocki A, Zdunek R, Amari S. Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization. in: 7th International Conference, ICA 2007. Berlin: Springer, 2007, 169~176.
[18] Ye Jieping. Generalized low rank approximations of matrices. Machine Learning, 2005, 61(1~3): 167~191.
[19] He X F, Cai D, Niyogi Partha. Tensor subspace analysis. in: NIPS, 2005: 1.
[20] Sun J T, Zeng H J, Liu H, et al. CubeSVD: a novel approach to personalized Web search. in: Proceedings of the 14th international conference on World Wide Web, 2005: 382~390.
[21] Harshman R A. Foundations of the PARAFAC procedure: models and conditions for an" explanatory" multimodal factor analysis. UCLA Working Papers in Phonetics, 1970, 16: 1~84.
[22] Harshman R A. PARAFAC2: Mathematical and technical notes. UCLA working papers in phonetics, 1972, 22: 30~44.
[23] ten B, Jos M. Simplicity and typical rank results for three-way arrays. Psychometrika, 2011, 76(1): 3~12.
[24] Carroll J D, Chang J J. Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 1970, 35(3): 283~319.
[25] Kolda T G, Bader B W. The TOPHITS model for higher-order web link analysis. in:Workshop on link analysis, counterterrorism and security, 2006: 26~29.
[26] Bader B W, Berry M W, Browne Murray. Discussion tracking in Enron email using PARAFAC. In Survey of Text Mining II. London: Springer, 2008, 147~163.
[27] Bro R, De J S. A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics, 1997, 11(5): 393~401.
[28] Friedlander M P, Hatz K. Computing non-negative tensor factorizations. Optimisation Methods and Software, 2008, 23(4): 631~647.
[29] Paatero P. A weighted non-negative least squares algorithm for three-way ‘PARAFAC’factor analysis. Chemometrics and Intelligent Laboratory Systems, 1997, 38(2): 223~242.
[30] Hazan T, Polak S, Shashua A. Sparse image coding using a 3D non-negative tensor factorization. in: Computer Vision, 2005, Tenth IEEE International Conference on, 2005: 50~57.
[31] Kong H, Teoh E K, Wang J G, et al. Two dimensional fisher discriminant analysis: Forget about small sample size problem. in: Proc. IEEE Intern. Conf. on Acoustics, Speech, and Signal Processing, 2005: 761~764.
[32] Yan SC, Xu D, Yang Q, et al. Multilinear discriminant analysis for face recognition. Image Processing, IEEE Transactions on, 2007, 16(1): 212~220.
[33] Tao D C, Li X L, Wu X D, et al. General tensor discriminant analysis and gabor features for gait recognition. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 2007, 29(10): 1700~1715.
[34] Jin Q Y, Huang Y Z, Wang C F. Modular discriminant analysis and its applications. Artificial Intelligence Review, 2013, 39(4): 285~303.
[35] Kotsia I, Patras I. Multiplicative update rules for multilinear support tensor machines. in: Pattern Recognition (ICPR), 2010 20th International Conference on, 2010: 33~36.
[36] Kotsia I, Guo W W, Patras I. Higher rank support tensor machines for visual recognition. Pattern Recognition, 2012, 45(12): 4192~4203.
[37] Yang X W, Chen B Q, Chen J. A tensor factorization based least squares support tensor machine for classification. In Advances in Neural Networks–ISNN 2013. Springer: 2013, 437~446.
[38] 曾奎, 何丽芳, 杨晓伟. 基于多线性主成分分析的支持高阶张量机. 南京大学学报(自然科学), 2014, 50(2): 219-227.

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