CUR matrix approximation: Difference between revisions

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The CUR matrix approximation is often {{citation needed|date=November 2012}} used in place of the low-rank approximation of the SVD in [[principal component analysis]]. The CUR is less accurate, but the columns of the matrix ''C'' are taken from ''A'' and the rows of ''R'' are taken from ''A''. In PCA, each column of ''A'' contains a data sample; thus, the matrix ''C'' is made of a subset of data samples. This is much easier to interpret than the SVD's left singular vectors, which represent the data in a rotated space. Similarly, the matrix ''R'' is made of a subset of variables measured for each data sample. This is easier to comprehend than the SVD's right singular vectors, which are another rotations of the data in space.

==Mathematical Definition==

Hamm and Huang<ref>Keaton Hamm and Longxiu Huang. Perspectives on CUR decompositions. Applied and Computational Harmonic Analysis, 48(3):1088–1099, 2020.</ref> gives the following theorem describing the basics of 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]].

In other words, if <math>L</math> has low rank, we can take a sub-matrix <math>U = L_{I,J}</math> of the same rank, together with some rows <math>R</math> and columns <math>C</math> of <math>L</math> and use them to reconstruct <math>L</math>.

==Algorithms==