Generalizing Dynamic Mode Decomposition: Balancing Accuracy and Expressiveness in Koopman Approximations

The paper’s core claims (ε-apart bound and relative RMS bound) are proved correctly and align with the candidate’s derivations. Theorem 6.1 (ε-apart) follows from the T-SSD construction and Step 7–8 (projection-difference and symmetric-intersection), and Theorem 6.2 (RMS ≤ ε) follows by rewriting the EDMD residual in terms of projector differences, consistent with the candidate’s projector-based proof sketch . However, for Theorem 6.3 and Proposition 6.5, the paper asserts that Koopman invariance implies R(D(X)Cmax) = R(D(Y)Cmax) and then deduces that Imax is preserved by T-SSD and that all eigenfunctions in span(D) are captured for any ε ∈ [0,1] . This equality can fail on sampled data in the presence of a nilpotent (e.g., λ=0) component: for an eigenfunction φ with λ=0 one may have φ(Y)=0 while φ(X)≠0, which puts φ(X) in the ±1-eigenspace of G0 and makes it unrecoverable for any ε<1. The model explicitly adds the needed nondegeneracy assumption (no nilpotent part, equivalently R(D(X)C*)=R(D(Y)C*)) and restricts (4) to λ≠0, thereby producing a correct argument where the paper’s proof omits a necessary hypothesis. The remainder of the paper (algorithm, ε-apart notion, EDMD special cases) is consistent and well grounded; e.g., the eigenvalues of the projection difference lie in [−1,1] as noted in the paper’s analysis of the ε=1 case , and the definition of ε-apart spaces matches the projector viewpoint used by the model .