The "novel information criterion" (NIC) algorithm was developed by Miao and Hua in 1998 for fast adaptive computation of the principal subspace of a vector sequence. The NIC algorithm is as efficient computationally as the PAST method, which was devised by Yang in 1995, and also has an attractive orthonormal property. Although all available evidence suggests that the NIC algorithm converges to the desired solution for any fixed leakage factor between zero and one, a complete proof (or disproof) has not been found, except for an arbitrarily small leakage factor. This paper presents this long-standing open problem with a discussion of what is known so far. The results shown in this paper provide a new insight into the orthonormal property of the NIC algorithm at convergence.

Oja's principal subspace algorithm is a well-known and powerful technique for learning and trackingprincipal information in time series. A thorough investigation of the convergence property of Oja'salgorithm is undertaken in this paper. The asymptotic convergence rates of the algorithm is discovered. The dependence of the algorithm on its initial weight matrix and the singularity of the data covariance matrix is comprehensively addressed

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.