Estimating the Largest Elements of a Matrix

We derive an algorithm for estimating the largest p ≥ 1 values aij or |aij | for an m×n matrix A, along with their locations in the matrix. The matrix is accessed using only matrix– vector or matrix–matrix products. For p = 1 the algorithm estimates the norm ‖A‖M := maxi,j |aij | or maxi,j aij . The algorithm is based on a power method for mixed subordinate matrix norms and iterates on n × t matrices, where t ≥ 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 matrix–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 matrix–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.