ON THE EMERGENCE OF NUMERICAL INSTABILITIES IN NEXT GENERATION RESERVOIR COMPUTING

The paper states as a formal Result that, for small time lag τ and step size h, the NGRC feature matrix Ψ is rank deficient for any delay dimension k>1, and its proof asserts linear dependence of certain columns; however, the provided argument actually shows only that two columns become O(h)-close via a mean-value/Euler step and then appeals to continuity of singular values to claim small σ_min, not σ_min=0. In short, the paper’s proof demonstrates ill-conditioning (near-dependence), not exact rank deficiency for any fixed h>0, while the model’s solution correctly quantifies σ_min(Ψ) ≤ C·τh and explains that exact dependence requires extra structure on F and is not generic. See Section 5.2 and the displayed equations and column constructions in the paper’s proof for details .