CUR matrix approximation: Difference between revisions

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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 the product ''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 {{citation needed|date=November 2012}} used in place of the low-rank approximation of the SVD in [[Principalprincipal componentscomponent 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==