Discrete empirical interpolation in the tensor t-product framework
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914978486812672 |
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| author | Chellappa, Sridhar Feng, Lihong Benner, Peter |
| author_facet | Chellappa, Sridhar Feng, Lihong Benner, Peter |
| contents | The discrete empirical interpolation method (DEIM) is a well-established approach, widely used for state reconstruction using sparse sensor/measurement data, nonlinear model reduction, and interpretable feature selection. We introduce the tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data. The proposed approach seeks to overcome one of the key drawbacks of DEIM, viz., the need for matricizing the data, which can distort any structural and/or geometric information. Our method leverages the recently developed tensor t-product algebra to avoid reshaping the data. In analogy with the standard DEIM, we formulate and solve a tensor-valued least-squares problem, whose solution is achieved through an interpolatory projection. We develop a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation. Using five different tensor-valued datasets, we numerically illustrate the better approximation properties of t-Q-DEIM and the significant computational cost reduction it offers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete empirical interpolation in the tensor t-product framework Chellappa, Sridhar Feng, Lihong Benner, Peter Numerical Analysis Computational Engineering, Finance, and Science Dynamical Systems The discrete empirical interpolation method (DEIM) is a well-established approach, widely used for state reconstruction using sparse sensor/measurement data, nonlinear model reduction, and interpretable feature selection. We introduce the tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data. The proposed approach seeks to overcome one of the key drawbacks of DEIM, viz., the need for matricizing the data, which can distort any structural and/or geometric information. Our method leverages the recently developed tensor t-product algebra to avoid reshaping the data. In analogy with the standard DEIM, we formulate and solve a tensor-valued least-squares problem, whose solution is achieved through an interpolatory projection. We develop a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation. Using five different tensor-valued datasets, we numerically illustrate the better approximation properties of t-Q-DEIM and the significant computational cost reduction it offers. |
| title | Discrete empirical interpolation in the tensor t-product framework |
| topic | Numerical Analysis Computational Engineering, Finance, and Science Dynamical Systems |
| url | https://arxiv.org/abs/2410.14519 |