Machine Learning Meliorates Computing and Robustness in Discrete Combinatorial Optimization Problems
Discrete combinatorial optimization problems in real world are typically defined via an ensemble of potentially high dimensional measurements pertaining to all subjects of a system under study. We point out that such a data ensemble in fact embeds with system's information content that is not d...
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| Veröffentlicht in: | Frontiers in applied mathematics and statistics Jg. 2 |
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Frontiers Media S.A
01.11.2016
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| Abstract | Discrete combinatorial optimization problems in real world are typically defined via an ensemble of potentially high dimensional measurements pertaining to all subjects of a system under study. We point out that such a data ensemble in fact embeds with system's information content that is not directly used in defining the combinatorial optimization problems. Can machine learning algorithms extract such information content and make combinatorial optimizing tasks more efficient? Would such algorithmic computations bring new perspectives into this classic topic of Applied Mathematics and Theoretical Computer Science? We show that answers to both questions are positive. One key reason is due to permutation invariance. That is, the data ensemble of subjects' measurement vectors is permutation invariant when it is represented through a subject-vs-measurement matrix. An unsupervised machine learning algorithm, called Data Mechanics (DM), is applied to find optimal permutations on row and column axes such that the permuted matrix reveals coupled deterministic and stochastic structures as the system's information content. The deterministic structures are shown to facilitate geometry-based divide-and-conquer scheme that helps optimizing task, while stochastic structures are used to generate an ensemble of mimicries retaining the deterministic structures, and then reveal the robustness pertaining to the original version of optimal solution. Two simulated systems, Assignment problem and Traveling Salesman problem, are considered. Beyond demonstrating computational advantages and intrinsic robustness in the two systems, we propose brand new robust optimal solutions. We believe such robust versions of optimal solutions are potentially more realistic and practical in real world settings. |
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| AbstractList | Discrete combinatorial optimization problems in real world are typically defined via an ensemble of potentially high dimensional measurements pertaining to all subjects of a system under study. We point out that such a data ensemble in fact embeds with system's information content that is not directly used in defining the combinatorial optimization problems. Can machine learning algorithms extract such information content and make combinatorial optimizing tasks more efficient? Would such algorithmic computations bring new perspectives into this classic topic of Applied Mathematics and Theoretical Computer Science? We show that answers to both questions are positive. One key reason is due to permutation invariance. That is, the data ensemble of subjects' measurement vectors is permutation invariant when it is represented through a subject-vs-measurement matrix. An unsupervised machine learning algorithm, called Data Mechanics (DM), is applied to find optimal permutations on row and column axes such that the permuted matrix reveals coupled deterministic and stochastic structures as the system's information content. The deterministic structures are shown to facilitate geometry-based divide-and-conquer scheme that helps optimizing task, while stochastic structures are used to generate an ensemble of mimicries retaining the deterministic structures, and then reveal the robustness pertaining to the original version of optimal solution. Two simulated systems, Assignment problem and Traveling Salesman problem, are considered. Beyond demonstrating computational advantages and intrinsic robustness in the two systems, we propose brand new robust optimal solutions. We believe such robust versions of optimal solutions are potentially more realistic and practical in real world settings. |
| Author | Hsieh, Fushing Fujii, Kevin Hsieh, Cho-Jui |
| Author_xml | – sequence: 1 givenname: Fushing surname: Hsieh fullname: Hsieh, Fushing – sequence: 2 givenname: Kevin surname: Fujii fullname: Fujii, Kevin – sequence: 3 givenname: Cho-Jui surname: Hsieh fullname: Hsieh, Cho-Jui |
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| Cites_doi | 10.1287/moor.2.3.209 10.1017/CBO9780511814068 10.1063/1.1703773 10.1142/0271 10.1007/s10955-014-1043-6 10.1103/PhysRevE.82.061110 10.1007/978-1-4757-3860-5 10.1137/0105003 10.1038/nphys2190 10.1007/s00453-009-9340-1 10.1198/016214504000001303 10.1126/science.220.4598.671 10.1371/journal.pone.0106154 10.1145/321105.321111 10.1016/j.ejor.2007.11.062 10.1090/S0273-0979-03-00975-3 10.5486/PMD.1959.6.3-4.12 10.1002/nav.3800020109 10.1109/iccvw.2009.5457690 10.1145/290179.290180 10.1137/1.9781611972238 10.1371/journal.pone.0056259 10.1017/S0963548310000039 10.1016/0304-3975(77)90012-3 10.3389/fams.2016.00009 10.1088/0305-4470/36/12/201 |
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| References | Papadimitriou (B5) 1977; 4 Fushing (B22) 2014; 9 Burkard (B3) 2012 Mémoli (B36) 2009 Li (B26) 1993 Chen (B32) 2005; 100 Rissanen (B25) 1989 Kapp (B15) 1977; 2 Fushing (B35) 2016; 2 Si (B29) 2014 Erdos (B17) 1959; 6 Arora (B6) 1998; 45 Bollob1s (B18) 2001 Fushing (B23) 2015; 12 Hsieh (B28) 2012 Kuhn (B1) 1955; 2 Mehta (B11) 2004 Parisi (B13) 1998 Lawler (B7) 1985 Dyson (B19) 1962; 3 Mackey (B30) 2011 Fushing (B20) 2010; 82 Diaconis (B9) 2003; 40 Kirkpatrick (B16) 1983; 220 Barvinok (B33) 2010; 19 Fushing (B34) 2014; 156 Applegate (B8) 2006 Bayati (B31) 2010; 58 Krokhmal (B14) 2009; 194 Munkres (B2) 1957; 5 Hsieh (B27) 2014 Mezard (B12) 1986 Fushing (B21) 2013; 8 Forrester (B10) 2003; 36 Crutchfield (B24) 2012; 8 Bellman (B4) 1962; 9 |
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