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Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods
Diego Olguín, Axel Osses, Héctor Ramírez
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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.
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