A new look at the power method for fast subspace tracking

A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively O(n(2)p), O(np(2)), and O(np) flops of computation at each iteration (update), where n. is the dimension of the vector sequence and p is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The O(np) implementation of the natural power method is shown to be superior to the O(np) equivalent of the Oja, PAST, and NIC methods. Like all power-based methods, the natural power method can be easily modified via subspace deflation to track the principal components and, hence, the rank of the principal subspace. (C)1999 Academic Press.