Literature survey on low rank approximation of matrices

Abstract

Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization, Interpolative decomposition, etc. are classical deterministic algorithms for low rank approximation. But these techniques are very expensive ( operations are required for matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n. In this article we review low rank approximation techniques briefly and give extensive references of many techniques.

Keywords:

• Singular value decomposition
• rank
• QR decomposition
• spectral norm
• Frobenius norm
• complexity
• interpolative decomposition
• subset selection
• randomized algorithm
• subsampling
• random projection
• cross/skeleton decomposition
• pseudoskeleton approximation
• pseudoinverse
• maximal volume
• adaptive cross approximation
• pivot

AMS Subject Classifications:

• 65F30
• 68W20
• 68W25

Acknowledgements

The authors wishes to thank Prof. Wolfgang Hackbusch for his constant support and encouragement during their stay at Max-Planck institute for Mathematics in the Sciences, Leipzig, Germany.

Notes

No potential conflict of interest was reported by the authors.

1 Different conditions can be considered.

2 .

3 In general the index at kth step should not be any index obtained at previous steps .

Additional information

Funding

This work was supported by National Board of Higher Mathematics, India [grant number 2/48(2)/2014/NBHM(R.P.)/R&D-II/32].