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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.19220 |
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| _version_ | 1866917654170697728 |
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| author | Chen, Yin-Jen Tang, Minh |
| author_facet | Chen, Yin-Jen Tang, Minh |
| contents | We study the matrix-variate regression problem $Y_i = \sum_{k} β_{1k} X_i β_{2k}^{\top} + E_i$ for $i=1,2\dots,n$ in the high dimensional regime wherein the response $Y_i$ are matrices whose dimensions $p_{1}\times p_{2}$ outgrow both the sample size $n$ and the dimensions $q_{1}\times q_{2}$ of the predictor variables $X_i$ i.e., $q_{1},q_{2} \ll n \ll p_{1},p_{2}$. We propose an estimation algorithm, termed KRO-PRO-FAC, for estimating the parameters $\{β_{1k}\} \subset \Re^{p_1 \times q_1}$ and $\{β_{2k}\} \subset \Re^{p_2 \times q_2}$ that utilizes the Kronecker product factorization and rearrangement operations from Van Loan and Pitsianis (1993). The KRO-PRO-FAC algorithm is computationally efficient as it does not require estimating the covariance between the entries of the $\{Y_i\}$. We establish perturbation bounds between $\hatβ_{1k} -β_{1k}$ and $\hatβ_{2k} - β_{2k}$ in spectral norm for the setting where either the rows of $E_i$ or the columns of $E_i$ are independent sub-Gaussian random vectors. Numerical studies on simulated and real data indicate that our procedure is competitive, in terms of both estimation error and predictive accuracy, compared to other existing methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19220 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regression for matrix-valued data via Kronecker products factorization Chen, Yin-Jen Tang, Minh Machine Learning We study the matrix-variate regression problem $Y_i = \sum_{k} β_{1k} X_i β_{2k}^{\top} + E_i$ for $i=1,2\dots,n$ in the high dimensional regime wherein the response $Y_i$ are matrices whose dimensions $p_{1}\times p_{2}$ outgrow both the sample size $n$ and the dimensions $q_{1}\times q_{2}$ of the predictor variables $X_i$ i.e., $q_{1},q_{2} \ll n \ll p_{1},p_{2}$. We propose an estimation algorithm, termed KRO-PRO-FAC, for estimating the parameters $\{β_{1k}\} \subset \Re^{p_1 \times q_1}$ and $\{β_{2k}\} \subset \Re^{p_2 \times q_2}$ that utilizes the Kronecker product factorization and rearrangement operations from Van Loan and Pitsianis (1993). The KRO-PRO-FAC algorithm is computationally efficient as it does not require estimating the covariance between the entries of the $\{Y_i\}$. We establish perturbation bounds between $\hatβ_{1k} -β_{1k}$ and $\hatβ_{2k} - β_{2k}$ in spectral norm for the setting where either the rows of $E_i$ or the columns of $E_i$ are independent sub-Gaussian random vectors. Numerical studies on simulated and real data indicate that our procedure is competitive, in terms of both estimation error and predictive accuracy, compared to other existing methods. |
| title | Regression for matrix-valued data via Kronecker products factorization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2404.19220 |