In answer to Satya's question 2:

The v's correspond to the eigenvectors in the standard formulation of PCA and the a's correspond to the coefficients these eigenvectors are weighted by -- theta_i = sum_i a_i * v_i. In the general formulation, the v_is are the principal components. I do not think that the a_i's and the v_i's are independent of each other, otherwise the algorithm would not have to iteratively update a's then v's. The hope with the iterative algorithm is that eventually the algorithm will converge to the optimal value for a and v.