Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods
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| Format: | Preprint |
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2025
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| _version_ | 1866918187647369216 |
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| author | Olguín, Diego Osses, Axel Ramírez, Héctor |
| author_facet | Olguín, Diego Osses, Axel Ramírez, Héctor |
| contents | In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is $O(N^{-1/2})$, with a constant related to the probability of success of the bound, given by Hoeffding's inequality, similar to other methodologies, such as Philipp et al. Furthermore, we propose a \textit{lifting back} operator to obtain trajectories generated by embedding the initial state and iterating a linear system in a higher dimension. This naturally yields an $O(N^{-1/2})$ error bound for mean trajectories. Finally, we show numerical results including an example of nonlinear system, exhibiting successful approximation with exponential decay faster than $-1/2$, as suggested by the theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods Olguín, Diego Osses, Axel Ramírez, Héctor Numerical Analysis Dynamical Systems 37M99, 47B32, 47B33, 65D1214 In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is $O(N^{-1/2})$, with a constant related to the probability of success of the bound, given by Hoeffding's inequality, similar to other methodologies, such as Philipp et al. Furthermore, we propose a \textit{lifting back} operator to obtain trajectories generated by embedding the initial state and iterating a linear system in a higher dimension. This naturally yields an $O(N^{-1/2})$ error bound for mean trajectories. Finally, we show numerical results including an example of nonlinear system, exhibiting successful approximation with exponential decay faster than $-1/2$, as suggested by the theoretical results. |
| title | Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods |
| topic | Numerical Analysis Dynamical Systems 37M99, 47B32, 47B33, 65D1214 |
| url | https://arxiv.org/abs/2506.09266 |