Bayesian Partial Reduced-Rank Regression
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913404532293632 |
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| author | Pintado, Maria F. Iacopini, Matteo Rossini, Luca Shestopaloff, Alexander Y. |
| author_facet | Pintado, Maria F. Iacopini, Matteo Rossini, Luca Shestopaloff, Alexander Y. |
| contents | Reduced-rank (RR) regression may be interpreted as a dimensionality reduction technique able to reveal complex relationships among the data parsimoniously. However, RR regression models typically overlook any potential group structure among the responses by assuming a low-rank structure on the coefficient matrix. To address this limitation, a Bayesian Partial RR (BPRR) regression is exploited, where the response vector and the coefficient matrix are partitioned into low- and full-rank sub-groups. As opposed to the literature, which assumes known group structure and rank, a novel strategy is introduced that treats them as unknown parameters to be estimated. The main contribution is two-fold: an approach to infer the low- and full-rank group memberships from the data is proposed, and then, conditionally on this allocation, the corresponding (reduced) rank is estimated. Both steps are carried out in a Bayesian approach, allowing for full uncertainty quantification and based on a partially collapsed Gibbs sampler. It relies on a Laplace approximation of the marginal likelihood and the Metropolized Shotgun Stochastic Search to estimate the group allocation efficiently. Applications to synthetic and real-world data reveal the potential of the proposed method to reveal hidden structures in the data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bayesian Partial Reduced-Rank Regression Pintado, Maria F. Iacopini, Matteo Rossini, Luca Shestopaloff, Alexander Y. Methodology Computation Reduced-rank (RR) regression may be interpreted as a dimensionality reduction technique able to reveal complex relationships among the data parsimoniously. However, RR regression models typically overlook any potential group structure among the responses by assuming a low-rank structure on the coefficient matrix. To address this limitation, a Bayesian Partial RR (BPRR) regression is exploited, where the response vector and the coefficient matrix are partitioned into low- and full-rank sub-groups. As opposed to the literature, which assumes known group structure and rank, a novel strategy is introduced that treats them as unknown parameters to be estimated. The main contribution is two-fold: an approach to infer the low- and full-rank group memberships from the data is proposed, and then, conditionally on this allocation, the corresponding (reduced) rank is estimated. Both steps are carried out in a Bayesian approach, allowing for full uncertainty quantification and based on a partially collapsed Gibbs sampler. It relies on a Laplace approximation of the marginal likelihood and the Metropolized Shotgun Stochastic Search to estimate the group allocation efficiently. Applications to synthetic and real-world data reveal the potential of the proposed method to reveal hidden structures in the data. |
| title | Bayesian Partial Reduced-Rank Regression |
| topic | Methodology Computation |
| url | https://arxiv.org/abs/2406.17444 |