Optimization flows landing on the Stiefel manifold

The paper proves (i) existence/uniqueness with N nonincreasing, (ii) logistic closed-form dynamics for S(t)=X(t)^T X(t) implying N→0 and X(t)→St(p,n), (iii) convergence of ω-limit points to the critical set C={X∈St: ψ(X)X=0}, and (iv) asymptotic stability of isolated local minima, using a compactness/contradiction argument for (iii) and semidefinite Lyapunov theory for (iv). See Proposition 5 for d/dt N=−λ||∇N||^2, Proposition 6 for χ̇=−2λχ(χ−I) with explicit eigenvalue formula, Theorem 7 for ω-limit ⊂ C, and Theorem 8 for asymptotic stability . The candidate solution establishes the same results via a different route: LaSalle’s invariance principle for (i)–(ii), a quantified “repeated drop” argument that controls the small normal term to f’s tangential descent for (iii), and robustness of asymptotic stability under vanishing perturbations plus a polar-decomposition decoupling for (iv). All steps align with the paper’s statements, and calculations (e.g., Ṡ=−2λ(S^2−S) and the eigenvalue logistic solution) match exactly. Minor differences are methodological, not substantive; the only paper issue is a small sign slip when referring to Ẋ=Λ in the stability proof, which does not affect correctness.