On residual bounds of the EDMD solution to the eigenvalue problem for the Koopman operator and backward shadowing stability of the EDMD/KMD

What the math gives to ML

The paper provides a constructive residual-to-backward-error pipeline for approximate Koopman eigenpairs: individual residuals quantify mode quality, while their aggregation yields an operator perturbation under which the computed spectrum is exact. This is transferable to learned latent dynamical systems, recurrent world models, and state-space sequence models, where a learned linear latent transition can be spectrally decomposed and monitored without relying only on rollout loss. The most practical adaptation is residual-based mode pruning and a backward-error penalty that suppresses spectrally plausible but dynamically invalid latent modes. These mechanisms should improve long-horizon stability and provide an inexpensive diagnostic for noisy or underspecified temporal data.