Compression of the Koopman matrix for nonlinear physical models via hierarchical clustering

The paper posits exact cluster-equality assumptions for the Koopman matrix (rows equal within each row-cluster and columns equal within each column-cluster), formalized in Eqs. (11)–(12) and then defines the compressed operator K′ by block averaging (Eq. (13)) and two compressed dictionaries, with ψ′A formed by row-cluster averaging (Eq. (21)) and ψ′B by column-cluster summation (Eq. (27)) . It then states K′ψ′B = ψ′A (Eq. (28)), introduces a recovery matrix R counting row-/column-cluster overlaps (Eq. (39)), and presents the square update laws RK′ψB = ψB(x_{t+1}) and K′RψA = ψA(x_{t+1}) (Eqs. (40)–(41)) . The candidate solution gives a direct algebraic proof of exactly these claims: (i) the two cluster-equality assumptions imply block constancy, so each K′ block average equals the common block value; this yields K′ψ′B = ψ′A; (ii) row-cluster equality implies ψA is constant on each row cluster, and with ψB(x_{t+1}) = ψA(x_t) and R_{ji} = |Cr,i ∩ Cc,j| one gets Rψ′A = ψ′B(x_{t+1}); composing gives the same two update laws. Thus, under the paper’s stated assumptions, both are correct and essentially the same proof, with the model providing a cleaner, fully general derivation. Note that the paper also acknowledges practical deviation from exact equalities and replaces some equalities with averages (e.g., Eq. (20)), but this does not affect the theorem-level statement under the exact cluster-equality hypothesis .