一种新的黎曼流形学习方法*

南京大学学报(自然科学版) ›› 2012, Vol. 48 ›› Issue (1): 108–114.

一种新的黎曼流形学习方法*

陈绍荣**王宏强，黎湘，夏胜平

出版日期:2015-05-21 发布日期:2015-05-21
作者简介: (国防科学技术大学四院一系电子科学与工程学院空间电子信息技术研究所，长沙，410073)
基金资助:
国家自然科学基金(60972114)

A novel algorithm for Riemannian manifold learning

Chen Shao-Rong ,Wang Hong一Qiang ,Li Xiang ,Xia Shenh一Ping

Online:2015-05-21 Published:2015-05-21
About author: (Research institute of Space Electronics information Technology, School of Electronic Science and Engineering,
National University of Defense Technology, Changsha, 410073，China)

摘要/Abstract

摘要： 本文提出一种新的黎曼流形学习为一法，在学习输人数据的低维流形结构的同时保持了输人数据与输出数据间的同态关系.该为一法的主要思想来源于曲线坐标系中协变坐标分量的几何表达，通过把
这种几何表达为一式转换应用于其有流形结构的输人数据集，能够分步、线性地直接计算出它们在嵌人空间中的低维坐标.在计算的过程中使用Dijkstra算法计算各点之间的最短距离，并使用了黎曼微分几何
中的一此基本概念.实验仿真分析结果表明了算法的有效性.

Abstract: We present a novel method for Ricmannian manifold learning, which identifies the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space with homomorphism
contained.The main idea is derived from the concept of covariant components in Curvilinear coordinates system. We translate this idea to a cloud of data points in order to calculate the coordinates of the points directly in a transparent
way. Our implementation currently uses Dijkstra’s algorithm for shortest paths in graphs and some basic concepts from Ricmannian differential geometry. Experimental results show the effectiveness of the algorithm.

陈绍荣**王宏强，黎湘，夏胜平. 一种新的黎曼流形学习方法*
[J]. 南京大学学报(自然科学版), 2012, 48(1): 108–114.

Chen Shao-Rong ,Wang Hong一Qiang ,Li Xiang ,Xia Shenh一Ping
. A novel algorithm for Riemannian manifold learning
[J]. Journal of Nanjing University(Natural Sciences), 2012, 48(1): 108–114.

参考文献

[1]Seung H，Lee D.The manifold ways of percep tion. Science，2000，290:2268一2269.
[2]um A，Wcstin C F, Hcrbcrthson M, et al. Fast manifold learning based on Ricmannian normal coordinates. Proceedings of the 14^th Scandinavian Conference on image Analysis，Joensuu，Finland，2005.
[3]Roweis S, Saul L. Nonlinear dimcnsionality re duction by locally linear embedding. Science 2000，290. 2323一2326.
[4]Belkin M，Niyogi P. Laplacian eigcnmaps for dimensionality reduction and data represcnta tion. Neural Computation, 2003，15(6) 1373一1396.
[5]Yang J，Li F X, Wang Y. A better scaled local tangent space alignment algorithm. Journal of Software, 2005 , 16 ( 9 ) : 1584- 1590.(杨剑，李伏欣，王压.一种改进的局部切空间排列算法.软件学报，2006, 16(9): 1584一1590).
[6]Tenncnbaum J，Silva V D, Langford J. A glob al geometric framework for nonlinear dimension ality reduction. Science, 2000，29 2319一2323.
[7]Min W L, Lu L, He X F. Locality pursuit em bedding. Pattern Recognition, 2004，37(4):781~788
[8]Zhang Z Y，Zha H Y. Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM Journal of Scientific Computing，2004，26(1):313一338.
[9]Arfken G. "Curvilinear coordinates"’ and "dif ferential vector operators". Mathematical Meth ods for Physicists.The 3^rd Edition. Orlando Academic Press, 1985，86一94.
[10]Lin T, Zha H B. Ricmannian manifold learn- ing. IEEE Transactions on Pattern Analysis and Machine intelligence, 2008，30:796一809.

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