An important matrix factorisation. TBC
Incremental updates / downdates
(Brand 2006, 2002; Bunch and Nielsen 1978; Gu and Eisenstat 1995, 1993; Sarwar et al. 2002; Zhang 2022).
as Frobenius minimiser
TODO.
For PCA
Incoming
Carlo Tomasi, elegantly pedagogic.
Avrim Blum on SVD
References
Bach, C, D. Ceglia, L. Song, and F. Duddeck. 2019. βRandomized Low-Rank Approximation Methods for Projection-Based Model Order Reduction of Large Nonlinear Dynamical Problems.β International Journal for Numerical Methods in Engineering 118 (4): 209β41.
Brand, Matthew. 2002. βIncremental Singular Value Decomposition of Uncertain Data with Missing Values.β In Computer Vision β ECCV 2002, edited by Anders Heyden, Gunnar Sparr, Mads Nielsen, and Peter Johansen, 2350:707β20. Berlin, Heidelberg: Springer Berlin Heidelberg.
βββ. 2006. βFast Low-Rank Modifications of the Thin Singular Value Decomposition.β Linear Algebra and Its Applications, Special Issue on Large Scale Linear and Nonlinear Eigenvalue Problems, 415 (1): 20β30.
Bunch, James R., and Christopher P. Nielsen. 1978. βUpdating the Singular Value Decomposition.β Numerische Mathematik 31 (2): 111β29.
Gu, Ming, and Stanley C. Eisenstat. 1993. βA Stable and Fast Algorithm for Updating the Singular Value Decomposition.β Citeseer.
βββ. 1995. βDowndating the Singular Value Decomposition.β SIAM Journal on Matrix Analysis and Applications 16 (3): 793β810.
Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp. 2010. βFinding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions.β arXiv.
Hastie, Trevor, Rahul Mazumder, Jason D. Lee, and Reza Zadeh. 2015. βMatrix Completion and Low-Rank SVD via Fast Alternating Least Squares.β In Journal of Machine Learning Research, 16:3367β3402.
Rabani, Eran, and Sivan Toledo. 2001. βOut-of-Core SVD and QR Decompositions.β In PPSC.
Sarwar, Badrul, George Karypis, Joseph Konstan, and John Riedl. 2002. βIncremental Singular Value Decomposition Algorithms for Highly Scalable Recommender Systems.β
Tropp, Joel A., Alp Yurtsever, Madeleine Udell, and Volkan Cevher. 2016. βRandomized Single-View Algorithms for Low-Rank Matrix Approximation.β arXiv:1609.00048 [Cs, Math, Stat], August.
Zhang, Yangwen. 2022. βAn Answer to an Open Question in the Incremental SVD.β arXiv.
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