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| Format: | Preprint |
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2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1710.04238 |
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| _version_ | 1866911828991279104 |
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| author | Tygert, Mark |
| author_facet | Tygert, Mark |
| contents | Linear least-squares regression with a "design" matrix A approximates a given matrix B via minimization of the spectral- or Frobenius-norm discrepancy ||AX-B|| over every conformingly sized matrix X. Another popular approximation is low-rank approximation via principal component analysis (PCA) -- which is essentially singular value decomposition (SVD) -- or interpolative decomposition (ID). Classically, PCA/SVD and ID operate solely with the matrix B being approximated, not supervised by any auxiliary matrix A. However, linear least-squares regression models can inform the ID, yielding regression-aware ID. As a bonus, this provides an interpretation as regression-aware PCA for a kind of canonical correlation analysis between A and B. The regression-aware decompositions effectively enable supervision to inform classical dimensionality reduction, which classically has been totally unsupervised. The regression-aware decompositions reveal the structure inherent in B that is relevant to regression against A. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_04238 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Regression-aware decompositions Tygert, Mark Methodology Machine Learning Numerical Analysis Linear least-squares regression with a "design" matrix A approximates a given matrix B via minimization of the spectral- or Frobenius-norm discrepancy ||AX-B|| over every conformingly sized matrix X. Another popular approximation is low-rank approximation via principal component analysis (PCA) -- which is essentially singular value decomposition (SVD) -- or interpolative decomposition (ID). Classically, PCA/SVD and ID operate solely with the matrix B being approximated, not supervised by any auxiliary matrix A. However, linear least-squares regression models can inform the ID, yielding regression-aware ID. As a bonus, this provides an interpretation as regression-aware PCA for a kind of canonical correlation analysis between A and B. The regression-aware decompositions effectively enable supervision to inform classical dimensionality reduction, which classically has been totally unsupervised. The regression-aware decompositions reveal the structure inherent in B that is relevant to regression against A. |
| title | Regression-aware decompositions |
| topic | Methodology Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/1710.04238 |