Convergence of Oja’s Online Principal Component Flow

The paper proves (i) global convergence of the flow Q̇=QΣ(A,Q) on O(n) to the finite set E of signed-permutation eigenbases (Theorem 5.1) using an LLG/gradient-flow framework with a strict Lyapunov function and LaSalle’s invariance principle, and (ii) a complete stable-manifold labeling via the recursively defined permutation σm together with orientation signs zm, yielding qσm(Q0)(t)→sgn(zm)em (Theorem 5.2), and (iii) exponential entrywise convergence rates under invertible leading principal minors (Theorem 5.3) with |qij(t)|^2−δij bounded by e−2νi∧jt, νk=min{λ1−λ2,…,λk−λk+1} (5.29–5.30) . The candidate solution proves the same results in the diagonal basis A=Λ and uses the continuous-QR/orthogonalization representation Y(t)=e^{tΛ}Q0=Q(t)R(t) to derive that Ṙ=(S−Σ(S))R is upper triangular, so Q(t) is the Q-factor of e^{tΛ}Q0; this is equivalent to the paper’s Cholesky-based identity Q(t)=e^{tA}Q0G(t) with G upper triangular and positive diagonal (R=G−1) . It then applies exterior-power (Plücker/wedge) asymptotics to identify the same σm and sgn(zm), and obtains the same exponential rates. Thus both are correct; the proofs are closely related but presented through different lenses (LLG/Cholesky/rank analysis in the paper vs. continuous QR/exterior algebra in the model).