## 36 Citations

### Pass-Efficient Randomized LU Algorithms for Computing Low-Rank Matrix Approximation

- Computer ScienceArXiv
- 2020

This paper proposes a novel randomized LU algorithm, called PowerLU, for the fixed low-rank approximation problem, and proposes a single-pass algorithm based on LU factorization, based on an efficient blocked adaptive rank determination Algorithm 4.1 proposed in this paper.

### Matrix decompositions using sub-Gaussian random matrices

- Computer ScienceInformation and Inference: A Journal of the IMA
- 2018

A new algorithm is presented, which achieves with high probability a rank-$r$ singular value decomposition (SVD) approximation of an $n \times n$ matrix and derive an error bound that does not depend on the first $ r$ singular values.

### Randomized Matrix Decompositions using

- Computer Science
- 2017

This work presents the R package rsvd, and provides a tutorial introduction to randomized matrix decompositions, showing the computational advantage over other methods implemented in R for approximating matrices with low-rank structure.

### Randomized Generalized Singular Value Decomposition

- Computer ScienceCommunications on Applied Mathematics and Computation
- 2020

This paper uses random projections to capture the most of the action of the matrices and proposes randomized algorithms for computing a low-rank approximation of the generalized singular value decomposition of two matrices.

### Randomized Matrix Decompositions using R

- Computer ScienceJournal of Statistical Software
- 2019

This work presents the R package rsvd, and provides a tutorial introduction to randomized matrix decompositions, showing the computational advantage over other methods implemented in R for approximating matrices with low-rank structure.

### Randomized algorithms for the low multilinear rank approximations of tensors

- Computer Science, MathematicsJ. Comput. Appl. Math.
- 2021

### Randomized algorithms for distributed computation of principal component analysis and singular value decomposition

- Computer ScienceAdv. Comput. Math.
- 2018

Carefully honed randomized algorithms yield results that are uniformly superior to those of the stock, deterministic implementations in Spark (the popular platform for distributed computation); in particular, whereas the stock software will without warning return left singular vectors that are far from numerically orthonormal, a significantly burnished randomized implementation generates left singular vector that are numerically Orthonormal to nearly the machine precision.

### An improved analysis and unified perspective on deterministic and randomized low rank matrix approximations

- Computer ScienceArXiv
- 2019

The established deterministic guarantees are combined with sketching ensembles satisfying Johnson-Lindenstrauss properties to present complete bounds and the factorization is shown to unify and generalize many past algorithms.

## References

SHOWING 1-10 OF 59 REFERENCES

### Randomized Algorithms for Low-Rank Matrix Factorizations: Sharp Performance Bounds

- Computer ScienceAlgorithmica
- 2014

A novel and rather intuitive analysis of the algorithm for approximating an input matrix with a low-rank element for dimensionality reduction is introduced, which allows it to derive sharp estimates and give new insights about its performance.

### Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

- Computer ScienceSIAM Rev.
- 2011

This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation, and presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions.

### Spectral Regularization Algorithms for Learning Large Incomplete Matrices

- Computer ScienceJ. Mach. Learn. Res.
- 2010

Using the nuclear norm as a regularizer, the algorithm Soft-Impute iteratively replaces the missing elements with those obtained from a soft-thresholded SVD in a sequence of regularized low-rank solutions for large-scale matrix completion problems.

### A randomized algorithm for the approximation of matrices

- Mathematics
- 2006

We introduce a randomized procedure that, given an m × n matrix A and a positive integer k, approximates A with a matrix Z of rank k. The algorithm relies on applying a structured l × m random matrix…

### On the existence and computation of rank-revealing LU factorizations

- Computer Science
- 2000

### A fast randomized algorithm for overdetermined linear least-squares regression

- Mathematics, Computer ScienceProceedings of the National Academy of Sciences
- 2008

A randomized algorithm for overdetermined linear least-squares regression based on QR-decompositions or bidiagonalization that computes an n × 1 vector x such that x minimizes the Euclidean norm ‖Ax − b‖ to relative precision ε.

### Smallest singular value of sparse random matrices

- Mathematics
- 2011

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition…

### Improved Matrix Algorithms via the Subsampled Randomized Hadamard Transform

- Computer Science, MathematicsSIAM J. Matrix Anal. Appl.
- 2013

This article addresses the efficacy, in the Frobenius and spectral norms, of an SRHT-based low-rank matrix approximation technique introduced by Woolfe, Liberty, Rohklin, and Tygert, and produces several results on matrix operations with SRHTs that may be of independent interest.

### A fast randomized algorithm for the approximation of matrices ✩

- Mathematics, Computer Science
- 2007

### Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix

- Computer ScienceSIAM J. Comput.
- 2006

Two simple and intuitive algorithms are presented which compute a description of a low-rank approximation of a singular value decomposition (SVD) to an matrix of rank not greater than a specified rank, and which are qualitatively faster than the SVD.