# Efficient nuclear norm approximation via the randomized UTV algorithm

## TL;DR

This paper presents a modified randUTV algorithm optimized for efficiently approximating the singular values of a matrix, enabling accurate nuclear norm computation with minimal additional cost.

## Contribution

It introduces a variant of randUTV tailored for singular value approximation and nuclear norm estimation, improving efficiency over the original full factorization approach.

## Key findings

- Effective approximation of singular values using the modified randUTV
- Accurate nuclear norm estimation with negligible additional cost
- Enhanced computational efficiency for Schatten-$p$ norms

## 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.