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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.18035 |
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| _version_ | 1866929692238413824 |
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| author | Armstrong, Robin Damle, Anil |
| author_facet | Armstrong, Robin Damle, Anil |
| contents | Column-pivoted QR (CPQR) factorization is a computational primitive used in numerous applications that require selecting a small set of ``representative'' columns from a much larger matrix. These include applications in spectral clustering, model-order reduction, low-rank approximation, and computational quantum chemistry, where the matrix being factorized has a moderate number of rows but an extremely large number of columns. We describe a modification of the Golub-Businger algorithm which, for many matrices of this type, can perform CPQR-based column selection much more efficiently. This algorithm, which we call CCEQR, is based on a three-step ``collect, commit, expand'' strategy that limits the number of columns being manipulated, while also transferring more computational effort from level-2 BLAS to level-3. Unlike most CPQR algorithms that exploit level-3 BLAS, CCEQR is deterministic, and provably recovers a column permutation equivalent to the one computed by the Golub-Businger algorithm. Tests on spectral clustering and Wannier basis localization problems demonstrate that on appropriately structured problems, CCEQR can significantly outperform GEQP3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Collect, Commit, Expand: Efficient CPQR-Based Column Selection for Extremely Wide Matrices Armstrong, Robin Damle, Anil Numerical Analysis Column-pivoted QR (CPQR) factorization is a computational primitive used in numerous applications that require selecting a small set of ``representative'' columns from a much larger matrix. These include applications in spectral clustering, model-order reduction, low-rank approximation, and computational quantum chemistry, where the matrix being factorized has a moderate number of rows but an extremely large number of columns. We describe a modification of the Golub-Businger algorithm which, for many matrices of this type, can perform CPQR-based column selection much more efficiently. This algorithm, which we call CCEQR, is based on a three-step ``collect, commit, expand'' strategy that limits the number of columns being manipulated, while also transferring more computational effort from level-2 BLAS to level-3. Unlike most CPQR algorithms that exploit level-3 BLAS, CCEQR is deterministic, and provably recovers a column permutation equivalent to the one computed by the Golub-Businger algorithm. Tests on spectral clustering and Wannier basis localization problems demonstrate that on appropriately structured problems, CCEQR can significantly outperform GEQP3. |
| title | Collect, Commit, Expand: Efficient CPQR-Based Column Selection for Extremely Wide Matrices |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2501.18035 |