Statistical complexity of dominant eigenvector calculation

We show that the number of arithmetic operations required to calculate a dominant -eigenvector of a real symmetric or complex Hermitian n × n matrix, when averaged over any density invariant under linear transformations that preserve the Frobenius norm, is bounded above by a polynomial in the size of the matrix. In fact, a specific upper bound is given in terms of n and . We also describe an estimate of the distance between an arbitrary complex n×m matrix and its rank one approximation. Introduction We present upper bounds on the statistical complexity of dominant -eigenvector calculation. We assume exact arithmetic throughout. See Blum et al. (1989), Smale (1985), and Traub et al. (1988) for a general discussion of this methodology. We restrict our attention to real symmetric and complex Hermitian matrices until Section 4. However, Definitions 2.1-2.4 apply to any matrix. Analogous results for stochastic matrices can be found in Wright (1989). Our definition of a dominant -eigenvector is a root error definition as opposed to a residual error definition: Definition. A vector v is a dominant -eigenvector of a matrix M if there exists a dominant eigenvector w of M with the property that ||v − w||2 ≤ ||v||2. Whenever dominant -eigenvectors are discussed, it is assumed that 0 < ≤ 1. This definition is better adapted to the exact arithmetic model than to models that include error. One of the referees described the limitations of the real number model and the chosen definition of the -eigenvector as follows. “[The author] assumes exact arithmetic, which makes his model easier to compute in than the usual floating point one. But his stopping criterion is quite strong, that the dominant eigenvector be computed with accuracy (in the norm sense) no matter how close λ1 and λ2 are. This is more than is asked in the usual situation, since if we compute with arithmetic with relative accuracy a, then perturbation theory implies that we expect uncertainty in the dominant eigenvector of about a/(1 − |λ2/λ1|). To meet the author’s stopping criterion, we would need at least that a < a/(1 − |λ2/λ1|). Thus the work would grow at least like log a log log a log log log a, (via Schönhage-Strassen multiplication) where log a grows like | log |+ log(λ2/λ1− 1). This is much more pessimistic than the author’s bound, which grows like log | log |, although the λ2/λ1 dependence is similar. “This may be a situation where the real number model studied by Smale and others is definitely ‘too strong’ to accurately model actual approximate computation, and this may be the most interesting conclusion.” The author will only add that the propagation of error inherent to the algorithm can be controlled by occasionally multiplying the iterates by the original matrix and then symmetrizing. A detailed error analysis is not include in this paper. We assume that the density chosen for the space of matrices is invariant under linear transformations that preserver the Frobenius norm. Examples of such densities are given in Section 1. However, as is implicit in Section 4 of Kostlan (1985), Wishart matrices would give similar results. ∗Department of Mathematics, University of Hawaii, Honolulu, Hawaii 97822