Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
© 2017 IEEE. In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the in...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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IEEE
2021
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| Online Access: | https://hdl.handle.net/1721.1/137768 |
| _version_ | 1826216573592928256 |
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| author | Cohen, Michael B. Madry, Aleksander Tsipras, Dimitris Vladu, Adrian |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Cohen, Michael B. Madry, Aleksander Tsipras, Dimitris Vladu, Adrian |
| author_sort | Cohen, Michael B. |
| collection | MIT |
| description | © 2017 IEEE. In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time \widetilde{O}(m\log \kappa \log^2 (1/∈)) where ∈ is the amount of error we are willing to tolerate. Here, \kappa represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever \kappa is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time \widetilde{O}(m^{3/2} \log (1/∈)).In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient. |
| first_indexed | 2024-09-23T16:49:27Z |
| format | Article |
| id | mit-1721.1/137768 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:49:27Z |
| publishDate | 2021 |
| publisher | IEEE |
| record_format | dspace |
| spelling | mit-1721.1/1377682023-08-11T17:11:11Z Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods Cohen, Michael B. Madry, Aleksander Tsipras, Dimitris Vladu, Adrian Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory © 2017 IEEE. In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time \widetilde{O}(m\log \kappa \log^2 (1/∈)) where ∈ is the amount of error we are willing to tolerate. Here, \kappa represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever \kappa is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time \widetilde{O}(m^{3/2} \log (1/∈)).In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient. 2021-11-08T18:39:01Z 2021-11-08T18:39:01Z 2017-10 2019-06-13T16:58:25Z Article http://purl.org/eprint/type/ConferencePaper https://hdl.handle.net/1721.1/137768 Cohen, Michael B., Madry, Aleksander, Tsipras, Dimitris and Vladu, Adrian. 2017. "Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods." en 10.1109/focs.2017.88 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf IEEE arXiv |
| spellingShingle | Cohen, Michael B. Madry, Aleksander Tsipras, Dimitris Vladu, Adrian Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods |
| title | Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods |
| title_full | Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods |
| title_fullStr | Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods |
| title_full_unstemmed | Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods |
| title_short | Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods |
| title_sort | matrix scaling and balancing via box constrained newton s method and interior point methods |
| url | https://hdl.handle.net/1721.1/137768 |
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