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Formally, a CUR matrix approximation of a matrix A is three matrices C, U, and R such that C is made from columns of A, R is made from rows of A, and that CUR closely approximates A. Usually the CUR is selected to be a [[Rank (linear algebra)|rank]]-k approximation, which means that C contains k columns of A, R contains k rows of A, and U is a k-by-k matrix. There are many possible CUR matrix approximations, and many CUR matrix approximations for a given rank.
The CUR matrix approximation is often used in place of the low-rank approximation of the SVD in [[Principle components 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.
==Algorithms==
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