Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods

In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is $O(N^{-1\mathrm{/}2})$, 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 $\mathit{\text{lifting}}\mathit{\text{back}}$ operator to obtain trajectories generated by embedding the initial state and iterating a linear system in a higher dimension. This naturally yields an $O(N^{-1\mathrm{/}2})$ error bound for mean trajectories. Finally, we show numerical results including an example of nonlinear system, exhibiting successful approximation with exponential decay faster than $-1\mathrm{/}2$, as suggested by the theoretical results.