Koopman Regularization

The paper formulates the constrained problem with O(m)=∑||J(m(x_i))p_i−1||^2 and a feasible barrier F(m) built from pairwise cosine constraints, choosing the stricter threshold cos^2θ≤1/N^2 (i.e., |cosθ|≤1/N) and noting that the classical sufficient condition −1/(N−1)<cosθ<1/(N−1) guarantees functional independence; see the definitions of O, F, the cosine measure, and the threshold choice in the core text and Appendix A . The candidate solution gives a clean Gram-matrix lower bound that (under |cosθ|≤1/N) yields strict positive definiteness, hence full rank of J(m)(x_i)—a correct but different proof route than the paper’s Appendix-A discussion (which analyzes the normalized Gram matrix and presents a determinant formula for the uniform-correlation case) . From O(m)=0 the candidate then recovers p_i=J(m)(x_i)^{-1}1, aligning exactly with the paper’s reconstruction formula p̂(x)=J(m)^{-1}1 when the Jacobian is full rank . For the K<N setting, the paper introduces the joint objective O(m,γ)=½∑||J(m)p−1||^2+½∑||p−J(m)^Tγ||^2 with a feasible region defined by the same cosine-style barrier (threshold 1/K^2) but does not analyze first-order conditions; the candidate correctly derives the normal equations JJ^Tγ=Jp and the projection interpretation, and sketches a plausible (interior-point) descent argument showing that at a local minimum p must lie in span{∇m_j}—an analysis step not provided in the paper . In sum, the paper’s claims for N functions and the candidate’s solution agree; the model supplies a tighter, explicit positive-definiteness bound under the paper’s stricter threshold and adds a reasonable K<N optimality argument the paper omits.