A DYNAMICAL SYSTEMS BASED FRAMEWORK FOR DIMENSION REDUCTION

The paper’s Theorem 4.1 claims that the encoder E(x)=Q h(T) is Lipschitz and sketches a Grönwall-based proof, but it replaces the key factor ||β|| by M (a bound on the dictionary entries) and applies Lipschitz/size bounds defined on K without ensuring trajectories remain in K; the theorem statement also asserts Lipschitzness for all x in R^d despite only assuming well-posedness for data points (and not global existence) . The candidate solution gives the standard, correct argument: f(h)=βΞ(h) is Lipschitz with constant ≤||β|| L_Ξ (with L_Ξ≤√(dn)L), so ∥h_1(T)−h_2(T)∥ ≤ e^{||β|| L_Ξ T}∥x_1−x_2∥ and hence ∥E(x_1)−E(x_2)∥ ≤ e^{||β|| L_Ξ T}∥x_1−x_2∥, further uniformized using ||β||≤b to yield C≤e^{b√(dn)L T}. This fixes both the constant and the missing domain condition (require Ξ to be Lipschitz on the set traversed by trajectories), aligning with Assumption 3.2 and B1(b) .