Adaptation dynamics of the spherical subspace tracker

L. Ljung´s (1977) method for analyzing recursive stochastic algorithms is used to formulate a projection operator ordinary differential equation (ODE). The ODE describes the expected convergence dynamics of a noniterative spherical subspace tracker. The subspace ODE is a Riccati equation defined over the manifold of rank r projection matrices in C^nxn. A Lyapunov function is defined that is shown to have global maximum and minimum at the signal and noise subspaces, respectively. By taking a derivative of the Lyapunov function along any trajectory, it is shown that the dynamics force all trajectories to converge to the signal subspace. If the sign of the derivative is changed, all trajectories will converge to the noise subspace