Structured Nuclear Norm Matrix Completion: Guaranteeing Exact Recovery via Block‐Column Scaling
ABSTRACT The goal of low‐rank matrix completion is to minimize the rank of a matrix while adhering to the constraint that known (non‐missing) elements are fixed in the approximation. Minimizing rank is a difficult, non‐convex, NP‐hard problem, often addressed by substituting rank with the nuclear no...
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| Published in | Numerical linear algebra with applications Vol. 32; no. 4 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken, USA
John Wiley & Sons, Inc
01.08.2025
Wiley Subscription Services, Inc |
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| Online Access | Get full text |
| ISSN | 1070-5325 1099-1506 1099-1506 |
| DOI | 10.1002/nla.70031 |
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| Abstract | ABSTRACT
The goal of low‐rank matrix completion is to minimize the rank of a matrix while adhering to the constraint that known (non‐missing) elements are fixed in the approximation. Minimizing rank is a difficult, non‐convex, NP‐hard problem, often addressed by substituting rank with the nuclear norm to achieve a convex relaxation. We focus on structured matrices for completion, where, in addition to the constraints described earlier, matrices also adhere to a predefined structure. We propose a technique that ensures the exact recovery of missing entries by minimizing the nuclear norm of a matrix where the non‐missing entries are first subject to block‐column scaling. We provide the proofs for exact recovery and propose a way for choosing the scaling parameter to ensure exact recovery. The method is demonstrated in several numerical examples, showing the usefulness of the proposed technique. |
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| AbstractList | ABSTRACT
The goal of low‐rank matrix completion is to minimize the rank of a matrix while adhering to the constraint that known (non‐missing) elements are fixed in the approximation. Minimizing rank is a difficult, non‐convex, NP‐hard problem, often addressed by substituting rank with the nuclear norm to achieve a convex relaxation. We focus on structured matrices for completion, where, in addition to the constraints described earlier, matrices also adhere to a predefined structure. We propose a technique that ensures the exact recovery of missing entries by minimizing the nuclear norm of a matrix where the non‐missing entries are first subject to block‐column scaling. We provide the proofs for exact recovery and propose a way for choosing the scaling parameter to ensure exact recovery. The method is demonstrated in several numerical examples, showing the usefulness of the proposed technique. The goal of low‐rank matrix completion is to minimize the rank of a matrix while adhering to the constraint that known (non‐missing) elements are fixed in the approximation. Minimizing rank is a difficult, non‐convex, NP‐hard problem, often addressed by substituting rank with the nuclear norm to achieve a convex relaxation. We focus on structured matrices for completion, where, in addition to the constraints described earlier, matrices also adhere to a predefined structure. We propose a technique that ensures the exact recovery of missing entries by minimizing the nuclear norm of a matrix where the non‐missing entries are first subject to block‐column scaling. We provide the proofs for exact recovery and propose a way for choosing the scaling parameter to ensure exact recovery. The method is demonstrated in several numerical examples, showing the usefulness of the proposed technique. |
| Author | Gillard, Jonathan Dreesen, Philippe Markovsky, Ivan Usevich, Konstantin |
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The goal of low‐rank matrix completion is to minimize the rank of a matrix while adhering to the constraint that known (non‐missing) elements are... The goal of low‐rank matrix completion is to minimize the rank of a matrix while adhering to the constraint that known (non‐missing) elements are fixed in the... |
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| SubjectTerms | Constraints exact recovery Matrix algebra matrix completion nuclear norm Recovery Scaling Structured matrices |
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| Title | Structured Nuclear Norm Matrix Completion: Guaranteeing Exact Recovery via Block‐Column Scaling |
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