Flexible Low-Rank Statistical Modeling with Missing Data and Side Information
We explore a general statistical framework for low-rank modeling of matrix-valued data, based on convex optimization with a generalized nuclear norm penalty. We study several related problems: the usual low-rank matrix completion problem with flexible loss functions arising from generalized linear m...
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| Format: | Article |
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Institute of Mathematical Statistics
2019
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| Online Access: | http://hdl.handle.net/1721.1/120549 https://orcid.org/0000-0003-1384-9743 |
| _version_ | 1826207372643663872 |
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| author | Fithian, William Mazumder, Rahul |
| author2 | Sloan School of Management |
| author_facet | Sloan School of Management Fithian, William Mazumder, Rahul |
| author_sort | Fithian, William |
| collection | MIT |
| description | We explore a general statistical framework for low-rank modeling of matrix-valued data, based on convex optimization with a generalized nuclear norm penalty. We study several related problems: the usual low-rank matrix completion problem with flexible loss functions arising from generalized linear models; reduced-rank regression and multi-task learning; and generalizations of both problems where side information about rows and columns is available, in the form of features or smoothing kernels. We show that our approach encompasses maximum a posteriori estimation arising from Bayesian hierarchical modeling with latent factors, and discuss ramifications of the missing-data mechanism in the context of matrix completion. While the above problems can be naturally posed as rank-constrained optimization problems, which are nonconvex and computationally difficult, we show how to relax them via generalized nuclear norm regularization to obtain convex optimization problems. We discuss algorithms drawing inspiration from modern convex optimization methods to address these large scale convex optimization computational tasks. Finally, we illustrate our flexible approach in problems arising in functional data reconstruction and ecological species distribution modeling. |
| first_indexed | 2024-09-23T13:48:19Z |
| format | Article |
| id | mit-1721.1/120549 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:48:19Z |
| publishDate | 2019 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1205492022-10-01T17:17:00Z Flexible Low-Rank Statistical Modeling with Missing Data and Side Information Fithian, William Mazumder, Rahul Sloan School of Management Mazumder, Rahul We explore a general statistical framework for low-rank modeling of matrix-valued data, based on convex optimization with a generalized nuclear norm penalty. We study several related problems: the usual low-rank matrix completion problem with flexible loss functions arising from generalized linear models; reduced-rank regression and multi-task learning; and generalizations of both problems where side information about rows and columns is available, in the form of features or smoothing kernels. We show that our approach encompasses maximum a posteriori estimation arising from Bayesian hierarchical modeling with latent factors, and discuss ramifications of the missing-data mechanism in the context of matrix completion. While the above problems can be naturally posed as rank-constrained optimization problems, which are nonconvex and computationally difficult, we show how to relax them via generalized nuclear norm regularization to obtain convex optimization problems. We discuss algorithms drawing inspiration from modern convex optimization methods to address these large scale convex optimization computational tasks. Finally, we illustrate our flexible approach in problems arising in functional data reconstruction and ecological species distribution modeling. United States. Office of Naval Research (Grant N000141512342) 2019-02-26T20:20:49Z 2019-02-26T20:20:49Z 2017-08 2019-02-25T21:17:42Z Article http://purl.org/eprint/type/JournalArticle 0883-4237 http://hdl.handle.net/1721.1/120549 Fithian, William and Rahul Mazumder. “Flexible Low-Rank Statistical Modeling with Missing Data and Side Information.” Statistical Science 33, 2 (May 2018): 238–260 © 2018 Institute of Mathematical Statistics https://orcid.org/0000-0003-1384-9743 http://dx.doi.org/10.1214/18-STS642 Statistical Science Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Fithian, William Mazumder, Rahul Flexible Low-Rank Statistical Modeling with Missing Data and Side Information |
| title | Flexible Low-Rank Statistical Modeling with Missing Data and Side Information |
| title_full | Flexible Low-Rank Statistical Modeling with Missing Data and Side Information |
| title_fullStr | Flexible Low-Rank Statistical Modeling with Missing Data and Side Information |
| title_full_unstemmed | Flexible Low-Rank Statistical Modeling with Missing Data and Side Information |
| title_short | Flexible Low-Rank Statistical Modeling with Missing Data and Side Information |
| title_sort | flexible low rank statistical modeling with missing data and side information |
| url | http://hdl.handle.net/1721.1/120549 https://orcid.org/0000-0003-1384-9743 |
| work_keys_str_mv | AT fithianwilliam flexiblelowrankstatisticalmodelingwithmissingdataandsideinformation AT mazumderrahul flexiblelowrankstatisticalmodelingwithmissingdataandsideinformation |