KMS Of Academy of mathematics and systems sciences, CAS
| ADAPTIVE LOW-NONNEGATIVE-RANK APPROXIMATION FOR STATE AGGREGATION OF MARKOV CHAINS | |
| Duan, Yaqi1; Wang, Mengdi1; Wen, Zaiwen2; Yuan, Yaxiang3 | |
| 2020 | |
| Source Publication | SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
 |
| ISSN | 0895-4798 |
| Volume | 41Issue:1Pages:244-278 |
| Abstract | 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. |
| Keyword | Markov chain state aggregation nonnegative matrix factorization atomic norm proximal alternating linearized minimization |
| DOI | 10.1137/18M1220790 |
| Indexed By | SCI |
| Language | 英语 |
| Funding Project | National Natural Science Foundation of China[11831002] ; National Natural Science Foundation of China[11421101] ; Beijing Academy of Artificial Intelligence (BAAI) |
| WOS Research Area | Mathematics |
| WOS Subject | Mathematics, Applied |
| WOS ID | WOS:000546980200011 |
| Publisher | SIAM PUBLICATIONS |
| Citation statistics | |
| Document Type | 期刊论文 |
| Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/51724 |
| Collection | 中国科学院数学与系统科学研究院 |
| Corresponding Author | Duan, Yaqi |
| Affiliation | 1.Princeton Univ, Dept Elect Engn, Princeton, NJ 08544 USA 2.Peking Univ, Natl Engn Lab Big Data Anal & Applicat, Beijing Int Ctr Math Res, Ctr Data Sci, Beijing, Peoples R China 3.Chinese Acad Sci, Acad Math & Syst Sci, State Key Lab Sci & Engn Comp, Beijing, Peoples R China |
| Recommended Citation 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. |
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