Randomized Rank-Revealing UZV Decomposition for Low-Rank Approximation of Matrices
For practitioners needing fast and accurate low-rank approximations of large dense matrices, this method offers a new decomposition with rank-revealing properties and linear complexity.
This paper introduces a randomized rank-revealing UZV decomposition (RRR-UZVD) for low-rank matrix approximation, achieving O(mnk) complexity and accurate rank revelation with small off-diagonal norms. Experiments on synthetic and real data (image reconstruction, robust PCA) demonstrate effectiveness.
Low-rank matrix approximation plays an increasingly important role in signal and image processing applications. This paper presents a new rank-revealing decomposition method called randomized rank-revealing UZV decomposition (RRR-UZVD). RRR-UZVD is powered by randomization to approximate a low-rank input matrix. Given a large and dense matrix ${\bf A} \in \mathbb R^{m \times n}$ whose numerical rank is $k$, where $k$ is much smaller than $m$ and $n$, RRR-UZVD constructs an approximation $\hat{\bf A}$ such as $\hat{\bf A}={\bf UZV}^T$, where ${\bf U}$ and ${\bf V}$ have orthonormal columns, the leading-diagonal block of ${\bf Z}$ reveals the rank of $\bf A$, and its off-diagonal blocks have small $\ell_2$-norms. RRR-UZVD is simple, accurate, and only requires a few passes through $\bf A$ with an arithmetic cost of $O(mnk)$ floating-point operations. To demonstrate the effectiveness of the proposed method, we conduct experiments on synthetic data, as well as real data in applications of image reconstruction and robust principal component analysis.