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. “Randomized Low-Rank Approximation Methods for Projection-Based Model Order Reduction of Large Nonlinear Dynamical Problems.” International Journal for Numerical Methods in Engineering 118 (4): 209–41.
Brand, Matthew. 2002. “Incremental Singular Value Decomposition of Uncertain Data with Missing Values.” In Computer Vision — ECCV 2002, edited by Anders Heyden, Gunnar Sparr, Mads Nielsen, and Peter Johansen, 2350:707–20. Berlin, Heidelberg: Springer Berlin Heidelberg.
———. 2006. “Fast Low-Rank Modifications of the Thin Singular Value Decomposition.” Linear Algebra and Its Applications, Special Issue on Large Scale Linear and Nonlinear Eigenvalue Problems, 415 (1): 20–30.
Bunch, James R., and Christopher P. Nielsen. 1978. “Updating the Singular Value Decomposition.” Numerische Mathematik 31 (2): 111–29.
Gu, Ming, and Stanley C. Eisenstat. 1993. “A Stable and Fast Algorithm for Updating the Singular Value Decomposition.” Citeseer.
———. 1995. “Downdating the Singular Value Decomposition.” SIAM Journal on Matrix Analysis and Applications 16 (3): 793–810.
Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp. 2010. “Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions.” arXiv.
Hastie, Trevor, Rahul Mazumder, Jason D. Lee, and Reza Zadeh. 2015. “Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares.” In Journal of Machine Learning Research, 16:3367–3402.
Rabani, Eran, and Sivan Toledo. 2001. “Out-of-Core SVD and QR Decompositions.” In PPSC.
Sarwar, Badrul, George Karypis, Joseph Konstan, and John Riedl. 2002. “Incremental Singular Value Decomposition Algorithms for Highly Scalable Recommender Systems.”
Tropp, Joel A., Alp Yurtsever, Madeleine Udell, and Volkan Cevher. 2016. “Randomized Single-View Algorithms for Low-Rank Matrix Approximation.” arXiv:1609.00048 [Cs, Math, Stat], August.
Zhang, Yangwen. 2022. “An Answer to an Open Question in the Incremental SVD.” arXiv.
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