ADAPTIVE LOW-NONNEGATIVE-RANK APPROXIMATION FOR STATE AGGREGATION OF MARKOV CHAINS

doi:10.1137/18M1220790

	ADAPTIVE LOW-NONNEGATIVE-RANK APPROXIMATION FOR STATE AGGREGATION OF MARKOV CHAINS
	Duan, Yaqi 1; Wang, Mengdi 1; Wen, Zaiwen 3; Yuan, Yaxiang 2
刊名	SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
	2020
卷号	41 期号:1 页码:244-278
关键词	Markov chain state aggregation nonnegative matrix factorization atomic norm proximal alternating linearized minimization
ISSN号	0895-4798
DOI	10.1137/18M1220790
英文摘要	This paper develops a low-nonnegative-rank approximation method to identify the state aggregation structure of a finite-state Markov chain under an assumption that the state space can be mapped into a handful of metastates. The number of metastates is characterized by the nonnegative rank of the Markov transition matrix. Motivated by the success of the nuclear norm relaxation in low-rank minimization problems, we propose an atomic regularizer as a convex surrogate for the nonnegative rank and formulate a convex optimization problem. Because the atomic regularizer itself is not computationally tractable, we instead solve a sequence of problems involving a nonnegative factorization of the Markov transition matrices by using the proximal alternating linearized minimization method. Two methods for adjusting the rank of factorization are developed so that local minima are escaped. One is to append an additional column to the factorized matrices, which can be interpreted as an approximation of a negative subgradient step. The other is to reduce redundant dimensions by means of linear combinations. Overall, the proposed algorithm very likely converges to the global solution. The efficiency and statistical properties of our approach are illustrated on synthetic data. We also apply our state aggregation algorithm on a Manhattan transportation data set and make extensive comparisons with an existing method.
资助项目	National Natural Science Foundation of China[11831002] ; National Natural Science Foundation of China[11421101] ; Beijing Academy of Artificial Intelligence (BAAI)
WOS研究方向	Mathematics
语种	英语
出版者	SIAM PUBLICATIONS
WOS记录号	WOS:000546980200011
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/51723]
专题	中国科学院数学与系统科学研究院
通讯作者	Duan, Yaqi
作者单位	1.Princeton Univ, Dept Elect Engn, Princeton, NJ 08544 USA 2.Chinese Acad Sci, Acad Math & Syst Sci, State Key Lab Sci & Engn Comp, Beijing, Peoples R China 3.Peking Univ, Natl Engn Lab Big Data Anal & Applicat, Beijing Int Ctr Math Res, Ctr Data Sci, Beijing, Peoples R China
推荐引用方式 GB/T 7714	Duan, Yaqi,Wang, Mengdi,Wen, Zaiwen,et al. ADAPTIVE LOW-NONNEGATIVE-RANK APPROXIMATION FOR STATE AGGREGATION OF MARKOV CHAINS[J]. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS,2020,41(1):244-278.
APA	Duan, Yaqi,Wang, Mengdi,Wen, Zaiwen,&Yuan, Yaxiang.(2020).ADAPTIVE LOW-NONNEGATIVE-RANK APPROXIMATION FOR STATE AGGREGATION OF MARKOV CHAINS.SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS,41(1),244-278.
MLA	Duan, Yaqi,et al."ADAPTIVE LOW-NONNEGATIVE-RANK APPROXIMATION FOR STATE AGGREGATION OF MARKOV CHAINS".SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 41.1(2020):244-278.