Efficient nuclear norm approximation via the randomized UTV algorithm

Abstract

The recently introduced algorithm randUTV provides a highly efficient technique for computing accurate approximations to all the singular values of a given matrix $A$. The original version of randUTV was designed to compute a full factorization of the matrix in the form $A = UTV^*$ where $U$ and $V$ are orthogonal matrices, and $T$ is upper triangular. The estimates to the singular values of $A$ appear along the diagonal of $T$. This manuscript describes how the randUTV algorithm can be modified when the only quantity of interest being sought is the vector of approximate singular values. The resulting method is particularly effective for computing the nuclear norm of $A$, or more generally, other Schatten-$p$ norms. The report also describes how to compute an estimate of the errors incurred, at essentially negligible cost.