On rank-2 Nonnegative Matrix Factorizations and their variants
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908469415641088 |
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| author | Lindy, Etna Noferini, Vanni Van Dooren, Paul |
| author_facet | Lindy, Etna Noferini, Vanni Van Dooren, Paul |
| contents | We consider the problem of finding the best nonnegative rank-2 approximation of an arbitrary nonnegative matrix. We first revisit the theory, including an explicit parametrization of all possible nonnegative factorizations of a nonnegative matrix of rank 2. Based on this result, we construct a cheaply computable (albeit suboptimal) nonnegative rank-2 approximation for an arbitrary nonnegative matrix input. This can then be used as a starting point for the Alternating Nonnegative Least Squares method to find a nearest approximate nonnegative rank-2 factorization of the input; heuristically, our newly proposed initial value results in both improved computational complexity and enhanced output quality. We provide extensive numerical experiments to support these claims. Motivated by graph-theoretical applications, we also study some variants of the problem, including matrices with symmetry constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On rank-2 Nonnegative Matrix Factorizations and their variants Lindy, Etna Noferini, Vanni Van Dooren, Paul Numerical Analysis 15B48, 65F99, 15A23, 15A18 We consider the problem of finding the best nonnegative rank-2 approximation of an arbitrary nonnegative matrix. We first revisit the theory, including an explicit parametrization of all possible nonnegative factorizations of a nonnegative matrix of rank 2. Based on this result, we construct a cheaply computable (albeit suboptimal) nonnegative rank-2 approximation for an arbitrary nonnegative matrix input. This can then be used as a starting point for the Alternating Nonnegative Least Squares method to find a nearest approximate nonnegative rank-2 factorization of the input; heuristically, our newly proposed initial value results in both improved computational complexity and enhanced output quality. We provide extensive numerical experiments to support these claims. Motivated by graph-theoretical applications, we also study some variants of the problem, including matrices with symmetry constraints. |
| title | On rank-2 Nonnegative Matrix Factorizations and their variants |
| topic | Numerical Analysis 15B48, 65F99, 15A23, 15A18 |
| url | https://arxiv.org/abs/2507.20612 |