Maboud F. Kaloorazi

Maboud F. Kaloorazi, Rodrigo C. de Lamare

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.

Maboud F. Kaloorazi, Jie Chen

Matrices with low numerical rank are omnipresent in many signal processing and data analysis applications. The pivoted QLP (p-QLP) algorithm constructs a highly accurate approximation to an input low-rank matrix. However, it is computationally prohibitive for large matrices. In this paper, we introduce a new algorithm termed Projection-based Partial QLP (PbP-QLP) that efficiently approximates the p-QLP with high accuracy. Fundamental in our work is the exploitation of randomization and in contrast to the p-QLP, PbP-QLP does not use the pivoting strategy. As such, PbP-QLP can harness modern computer architectures, even better than competing randomized algorithms. The efficiency and effectiveness of our proposed PbP-QLP algorithm are investigated through various classes of synthetic and real-world data matrices.

M. Kaloorazi, R. C. de Lamare

Low-rank matrix approximations play a fundamental role in numerical linear algebra and signal processing applications. This paper introduces a novel rank-revealing matrix decomposition algorithm termed Compressed Randomized UTV (CoR-UTV) decomposition along with a CoR-UTV variant aided by the power method technique. CoR-UTV is primarily developed to compute an approximation to a low-rank input matrix by making use of random sampling schemes. Given a large and dense matrix of size $m\times n$ with numerical rank $k$, where $k \ll \text{min} \{m,n\}$, CoR-UTV requires a few passes over the data, and runs in $O(mnk)$ floating-point operations. Furthermore, CoR-UTV can exploit modern computational platforms and, consequently, can be optimized for maximum efficiency. CoR-UTV is simple and accurate, and outperforms reported alternative methods in terms of efficiency and accuracy. Simulations with synthetic data as well as real data in image reconstruction and robust principal component analysis applications support our claims.