Low-Rank Approximation of a Matrix: Novel Insights, New Progress, and Extensions
Empirical performance of the celebrated algorithms for low-rank approximation of a matrix by means of random sampling has been consistently efficient in various studies with various sparse and structured multipliers, but so far formal support for this empirical observation has been missing. Our new insight into this subject enables us to provide such an elusive formal support. Furthermore, our approach promises significant acceleration of the known algorithms by means of sampling with more efficient sparse and structured multipliers. It should also lead to enhanced performance of other fundamental matrix algorithms. Our formal results and our initial numerical tests are in good accordance with each other, and we have already extended our progress to the acceleration of the Fast Multipole Method and the Conjugate Gradient algorithms.
KeywordsLow-rank approximation of a matrix Random sampling Derandomization Fast multipole method Conjugate gradient algorithms
Our research has been supported by NSF Grant CCF-1116736 and PSC CUNY Awards 67699-00 45 and 68862-00 46. We are also grateful to the reviewers for valuable comments.
- 3.Barba, L.A., Yokota, R.: How will the fast multipole method fare in exascale era? SIAM News 46(6), 1–3 (2013)Google Scholar
- 10.Eidelman, Y., Gohberg, I., Haimovici, I.: Separable Type Representations of Matrices and Fast Algorithms Volume 1: Basics Completion Problems. Multiplication and Inversion Algorithms, Volume 2: Eigenvalue Methods. Birkhauser, Basel (2013)Google Scholar
- 12.Frieze, A., Kannan, R., Vempala, S.: Fast Monte-Carlo algorithms for finding low-rank approximations. J. ACM 51, 1025–1041 (2004). (Proceedings version in 39th FOCS, pp. 370–378. IEEE Computer Society Press (1998))Google Scholar
- 20.Mahoney, M.W.: Randomized algorithms for matrices and data. Found. Trends Mach. Learn. 3(2) (2011). (In: Way, M.J., et al. (eds.) Advances in Machine Learning and Data Mining for Astronomy (abridged version), pp. 647-672. NOW Publishers (2012))Google Scholar
- 23.Pan, V.Y., Zhao, L.: Low-rank approximation of a matrix: novel insights, new progress, and extensions. arXiv:1510.06142 [math.NA]. Submitted on 21 October 2015, Revised April 2016