CUR matrix approximation: Difference between revisions

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==Mathematical definition==

Hamm<ref>Keaton Hamm and Longxiu Huang. Perspectives on CUR decompositions. Applied and Computational Harmonic Analysis, 48(3):1088–1099, 2020.</ref> and Aldroubi et al.<ref>@article{aldroubi2019cur, title={CUR decompositions, similarity matrices, and subspace clustering}, author={Aldroubi, Akram and Hamm, Keaton and Koku, Ahmet Bugra and Sekmen, Ali}. CUR decompositions, similarity matrices, journal={and subspace clustering. Frontiers in Applied Mathematics and Statistics}, volume={ 4}, pages={65}, year={2019}, publisher={Frontiers Media SA}</ref> describe the following theorem, which outlines a CUR decomposition of a matrix <math>L</math> with rank <math>r</math>:

Theorem: Consider row and column indices <math>I, J \subseteq [n]</math> with <math>|I|, |J| \ge r</math>. Denote submatrices <math>C = L_{:,J},</math> <math>U = L_{I,J}</math> and <math>R = L_{I,:}</math>. If rank(<math>U</math>) = rank(<math>L</math>), then <math>L = CU^+R</math>, where <math>(\cdot)^+</math> denotes the [[Moore–Penrose pseudoinverse]].