Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds

Abstract

Given a real matrix $\mathbf{A}$ with $n$ columns, the problem is to approximate the Gram product $\mathbf{A}\mathbf{A}^T$ by $c\ll n$ weighted outer products of columns of $\mathbf{A}$. Necessary and sufficient conditions for the exact computation of $\mathbf{A}\mathbf{A}^T$ (in exact arithmetic) from $c\geq \mathrm{rank}(\mathbf{A})$ columns depend on the right singular vector matrix of $\mathbf{A}$. For a Monte Carlo matrix multiplication algorithm by Drineas et al. that samples outer products, we present probabilistic bounds for the two-norm relative error due to randomization. The bounds depend on the stable rank or the rank of $\mathbf{A}$, but not on the matrix dimensions. Numerical experiments illustrate that the bounds are informative, even for stringent success probabilities and matrices of small dimension. We also derive bounds for the smallest singular value and the condition number of matrices obtained by sampling rows from orthonormal matrices.