Adaptive finite element type decomposition of Gaussian processes
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909628291350528 |
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| author | Kim, Jaehoan Bhattacharya, Anirban Pati, Debdeep |
| author_facet | Kim, Jaehoan Bhattacharya, Anirban Pati, Debdeep |
| contents | In this paper, we investigate a class of approximate Gaussian processes (GP) obtained by taking a linear combination of compactly supported basis functions with the basis coefficients endowed with a dependent Gaussian prior distribution. This general class includes a popular approach that uses a finite element approximation of the stochastic partial differential equation (SPDE) associated with Matérn GP. We explored another scalable alternative popularly used in the computer emulation literature where the basis coefficients at a lattice are drawn from a Gaussian process with an inverse-Gamma bandwidth. For both approaches, we study concentration rates of the posterior distribution. We demonstrated that the SPDE associated approach with a fixed smoothness parameter leads to a suboptimal rate despite how the number of basis functions and bandwidth are chosen when the underlying true function is sufficiently smooth. On the flip side, we showed that the later approach is rate-optimal adaptively over all smoothness levels of the underlying true function if an appropriate prior is placed on the number of basis functions. Efficient computational strategies are developed and numerics are provided to illustrate the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive finite element type decomposition of Gaussian processes Kim, Jaehoan Bhattacharya, Anirban Pati, Debdeep Statistics Theory Methodology Machine Learning In this paper, we investigate a class of approximate Gaussian processes (GP) obtained by taking a linear combination of compactly supported basis functions with the basis coefficients endowed with a dependent Gaussian prior distribution. This general class includes a popular approach that uses a finite element approximation of the stochastic partial differential equation (SPDE) associated with Matérn GP. We explored another scalable alternative popularly used in the computer emulation literature where the basis coefficients at a lattice are drawn from a Gaussian process with an inverse-Gamma bandwidth. For both approaches, we study concentration rates of the posterior distribution. We demonstrated that the SPDE associated approach with a fixed smoothness parameter leads to a suboptimal rate despite how the number of basis functions and bandwidth are chosen when the underlying true function is sufficiently smooth. On the flip side, we showed that the later approach is rate-optimal adaptively over all smoothness levels of the underlying true function if an appropriate prior is placed on the number of basis functions. Efficient computational strategies are developed and numerics are provided to illustrate the theoretical results. |
| title | Adaptive finite element type decomposition of Gaussian processes |
| topic | Statistics Theory Methodology Machine Learning |
| url | https://arxiv.org/abs/2505.24066 |