A Fast Randomized Algorithm for computing the Null Space

@article{Nakatsukasa2022AFR,
  title={A Fast Randomized Algorithm for computing the Null Space},
  author={Yuji Nakatsukasa and Taejung Park},
  journal={ArXiv},
  year={2022},
  volume={abs/2206.00975}
}
Randomized algorithms in numerical linear algebra can be fast, scalable and robust. This paper examines the effect of sketching on the right singular vectors corresponding to the smallest singular values of a tall-skinny matrix. We devise a fast algorithm for finding the trailing right singular vectors using randomization and examine the quality of the solution using multiplicative perturbation theory. For an m × n ( m ≥ n ) matrix, the algorithm runs with complexity O ( mn log n + n 3 ) which is… 
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References

SHOWING 1-10 OF 32 REFERENCES

Hashing embeddings of optimal dimension, with applications to linear least squares

Subspace embedding properties for s-hashing sketching matrices, with s ≥ 1, that are optimal in the projection dimension m of the sketch are presented, namely, m = O(d), where d is the dimension of the subspace.

Randomized numerical linear algebra: Foundations and algorithms

This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problems and treats both the theoretical foundations of the subject and practical computational issues.

Practical Sketching Algorithms for Low-Rank Matrix Approximation

A suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image, or sketch, of the matrix that can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximation with a user-specified rank.

Relative Perturbation Theory: II. Eigenspace and Singular Subspace Variations

It is proved that under these kinds of perturbations, the changes of invariant subspaces are proportional to the reciprocals of relative gaps between subset of spectra or subsets of singular values.

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

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.

The AAA Algorithm for Rational Approximation

We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of MATLAB and requiring no user input parameters. Even on a disk o...

DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES

In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of…

Sketching as a Tool for Numerical Linear Algebra

This survey highlights the recent advances in algorithms for numericallinear algebra that have come from the technique of linear sketching, and considers least squares as well as robust regression problems, low rank approximation, and graph sparsification.

Sketched SVD: Recovering Spectral Features from Compressive Measurements

A streaming data model in which n sensors observe individual streams of data, presented in a turnstile model, is considered to analyze the singular value decomposition (SVD) of the matrix of data defined implicitly by the stream of updates and to obtain a streaming graph algorithm to approximate the eigenvalues and eigenvectors of the graph Laplacian in the case where the graph has low rank.

Local operator theory, random matrices and Springer Nature 2021 LTEX template A Fast Randomized Algorithm for computing the Null Space 25 Banach spaces

  • Handbook of the geometry of Banach spaces,
  • 2001