Dimension-independent eigenvalue approximation for sparse matrices

Abstract

Many applications in science and engineering require solving sparse Hermitian eigenvalue problems that are so large that even algorithms whose runtime scales linearly with the matrix dimension are unacceptably slow. Responding to this challenge, there is a growing body of work on sparsified iterative methods and submatrix selection methods that often perform well in practice. However, theoretical understanding of these methods is limited. In this paper, we take a step toward better understanding sublinear-time eigenvalue computation by analyzing the sparsified power method. We prove that the method approximates a leading eigenpair to a prescribed accuracy in a number of iterations comparable to the standard power method. The required number of retained nonzeros depends explicitly on the target accuracy, the initial overlap with the leading eigenvector, the compressibility of the leading eigenvector, and the relative spectral gap. When these quantities and the column sparsity are independent of dimension, the runtime is independent of dimension. In contrast, we show that a comparable worst-case guarantee cannot hold for a prototypical sparsified submatrix selection method. Submatrix selection is the approach of choice in many quantum chemistry applications. However, we show that, even with substantial initial eigenvector overlap, a single iteration of the method can produce a vector that is orthogonal to the leading eigenvector, after which the method cannot recover. Nevertheless, experiments on matrices from the SuiteSparse matrix collection and quantum chemistry problems illustrate the practical effectiveness of submatrix selection methods in a variety of domains despite these worst-case limitations.