Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction

doi:10.1007/s10107-016-1090-7

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	Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction
	Ding, Chao1 ; Qi, Hou-Duo 2
刊名	MATHEMATICAL PROGRAMMING
	2017-07-01
卷号	164 期号:1-2 页码:341-381
关键词	Euclidean distance matrix Convex matrix optimization Multidimensional scaling Nonlinear dimensionality reduction Low-rank matrix Error bounds Random graph models
ISSN号	0025-5610
DOI	10.1007/s10107-016-1090-7
英文摘要	Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP) to reconstruct such faithful representations. While those SDP models are capable of producing high quality configuration numerically, they suffer two major drawbacks. One is that there exist no theoretically guaranteed bounds on the quality of the configuration. The other is that they are slow in computation when the data points are beyond moderate size. In this paper, we propose a convex optimization model of Euclidean distance matrices. We establish a non-asymptotic error bound for the random graph model with sub-Gaussian noise, and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low-rank matrix up to a logarithmic factor. Our results partially explain why MVU and MVE often work well. Moreover, the convex optimization model can be efficiently solved by a recently proposed 3-block alternating direction method of multipliers. Numerical experiments show that the model can produce configurations of high quality on large data points that the SDP approach would struggle to cope with.
资助项目	Engineering and Physical Science Research Council (UK)[EP/K007645/1] ; National Natural Science Foundation of China[11671387]
WOS研究方向	Computer Science ; Operations Research & Management Science ; Mathematics
语种	英语
出版者	SPRINGER HEIDELBERG
WOS记录号	WOS:000403450600014
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/25684]
专题	应用数学研究所
通讯作者	Qi, Hou-Duo
作者单位	1.Chinese Acad Sci, Inst Appl Math, Beijing, Peoples R China 2.Univ Southampton, Sch Math, Southampton SO17 1BJ, Hants, England
推荐引用方式 GB/T 7714	Ding, Chao,Qi, Hou-Duo. Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction[J]. MATHEMATICAL PROGRAMMING,2017,164(1-2):341-381.
APA	Ding, Chao,&Qi, Hou-Duo.(2017).Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction.MATHEMATICAL PROGRAMMING,164(1-2),341-381.
MLA	Ding, Chao,et al."Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction".MATHEMATICAL PROGRAMMING 164.1-2(2017):341-381.