Estimating the Largest Elements of a Matrix

We derive an algorithm for estimating the largest p A�¢A¢Â�°A�Â¥ 1 values a ij or |a ij | for an m A�Â�A¢Â�Â� n matrix A, along with their locations in the matrix. The matrix is accessed using only matrixA�¢A¢Â�¬A¢Â�Â� vector or matrixA�¢A¢Â�¬A¢Â�Â�matrix products. For p = 1 the algorithm estimates the norm A M := max i,j |a ij | or max i,j a ij . The algorithm is based on a power method for mixed subordinate matrix norms and iterates on n A�Â�A¢Â�Â� t matrices, where t A�¢A¢Â�°A�Â¥ p is a parameter. For p = t = 1 we show that the algorithm is essentially equivalent to rook pivoting in Gaussian elimination; we also obtain a bound for the expected number of matrixA�¢A¢Â�¬A¢Â�Â�vector products for random matrices and give a class of counter-examples. Our numerical experiments show that for p = 1 the algorithm usually converges in just two iterations, requiring the equivalent of 4t matrixA�¢A¢Â�¬A¢Â�Â�vector products, and for t = 2 the algorithm already provides excellent estimates that are usually within a factor 2 of the largest element and frequently exact. For p > 1 we incorporate deflation to improve the performance of the algorithm. Experiments on real-life datasets show that the algorithm is highly effective in practice.