<p>There is a paper called <a href="https://arxiv.org/pdf/0909.4061.pdf" rel="nofollow noreferrer">Finding Structure in Randomness</a> that address some points about all of these decompositions as well as the SVD which would be covered in <a href="http://bookstore.siam.org/ot50/" rel="nofollow noreferrer">Trefethan and Ba</a>u .  </p>

<ol>
<li>The interpolative decomposition is used in different places. A paper that explores it is <a href="https://arxiv.org/pdf/1412.8447.pdf" rel="nofollow noreferrer">here.</a></li>
<li>The U,V are unitary matrices. C is a matrix containing a subset of the columns of A, R a subset of the rows. </li>
</ol>
How we can use CUR decomposition in place of SVD decomposition?
<p>I have understood how <a href="https://en.wikipedia.org/wiki/CUR_matrix_approximation" rel="nofollow noreferrer">CUR</a> and <a href="https://en.wikipedia.org/wiki/Singular-value_decomposition" rel="nofollow noreferrer">SVD</a> works, but not able to understand,<br></p>

<ol>
<li>How we can use CUR in place of SVD decomposition?</li>
<li>Does C and R matrices in CUR follow the same properties as that of U and V matrices in SVD decomposition?</li>
</ol>

<p>If we want to reduce the dimension of original matrix say from n to k, which matrix of CUR we can use to project original matrix, so that we will get k-dimensional data points.</p>