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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/2507.20686 |
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| _version_ | 1866911132828041216 |
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| author | Xue, Feng Zhang, Hui |
| author_facet | Xue, Feng Zhang, Hui |
| contents | We investigate the solution properties of the regularized least-squares problem.
Using a subspace decomposition technique, we derive expressions for the solution set in terms of the
conjugate function, from which various properties, including existence, compactness and uniqueness,
can then be easily analyzed. A key distinction of our approach from existing works is the separate
treatment of existence and compactness. We unify many existing results based on recession cones and
sublevel sets, and link them to our findings by connecting the recession function with the recession
cone of the subdifferential of the conjugate function. In particular, the concept of restricted coercivity
is developed and discussed in various aspects. The associated linearly constrained counterpart is
discussed in a similar manner. Its connections to regularized least-squares are further established via
the exactness of infimal postcomposition. Our results are supported by numerous examples, among
which the geometric interpretation of the lasso solution deserves further investigations in near future. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20686 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subspace decomposition in regularized least-squares: solution properties, restricted coercivity and beyond Xue, Feng Zhang, Hui Optimization and Control 47H05, 49M29, 49M27, 90C25 We investigate the solution properties of the regularized least-squares problem. Using a subspace decomposition technique, we derive expressions for the solution set in terms of the conjugate function, from which various properties, including existence, compactness and uniqueness, can then be easily analyzed. A key distinction of our approach from existing works is the separate treatment of existence and compactness. We unify many existing results based on recession cones and sublevel sets, and link them to our findings by connecting the recession function with the recession cone of the subdifferential of the conjugate function. In particular, the concept of restricted coercivity is developed and discussed in various aspects. The associated linearly constrained counterpart is discussed in a similar manner. Its connections to regularized least-squares are further established via the exactness of infimal postcomposition. Our results are supported by numerous examples, among which the geometric interpretation of the lasso solution deserves further investigations in near future. |
| title | Subspace decomposition in regularized least-squares: solution properties, restricted coercivity and beyond |
| topic | Optimization and Control 47H05, 49M29, 49M27, 90C25 |
| url | https://arxiv.org/abs/2507.20686 |