MODULARIZED DATA-DRIVEN APPROXIMATION OF THE KOOPMAN OPERATOR AND GENERATOR

The paper’s Proposition 4.4 proves a finite-data, high-probability estimation error bound for mgEDMD under the weak interconnection Assumption 4.3 by (i) invoking the EDMD sampling lemma (Lemma 2.1) to bound local generator-estimation errors, (ii) using the local propagation inequality (4.3) and summing it to obtain (4.6), and (iii) applying a probability addition rule to ensure that both the direct terms and interconnection gains are simultaneously small with high probability, yielding P(||x−x̂||_∞ ≤ ε) ≥ δ for suitably large, per-subsystem sample sizes; see the model definitions (4.1)–(4.2), the weak-interconnection condition (4.5), and the proof steps (4.6)–(4.7) . The candidate solution follows the same architecture and logic: it uses Lemma 2.1 to pick sample sizes via a union bound, rewrites (4.3) as a vector inequality e ≤ a + Be, enforces a small-gain margin (column sums < 1) ensured near zero-estimation error by continuity/monotonicity of Eij, and then deduces ||e||1 ≤ ||a||1/(1−||B||1), selecting radii so that the target ε is met. This is mathematically equivalent in spirit to the paper’s summation-based small-gain argument, just expressed in matrix-norm form. Both arguments explicitly treat estimation error (the paper states projection error is deferred) , and both use the uniform-in-time one-norm definition ||·||∞ over [0,T] . Hence, the paper and the model solution agree on assumptions, structure, and conclusion, with only stylistic differences in how the small-gain step is written.