How we can use CUR decomposition in place of SVD decomposition?

Question

I have understood how CUR and SVD works, but not able to understand,

1. How we can use CUR in place of SVD decomposition?
2. Does C and R matrices in CUR follow the same properties as that of U and V matrices in SVD decomposition?

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.

Answer 1

There is a paper called Finding Structure in Randomness that address some points about all of these decompositions as well as the SVD which would be covered in Trefethan and Bau .

1. The interpolative decomposition is used in different places. A paper that explores it is here.
2. The U,V are unitary matrices. C is a matrix containing a subset of the columns of A, R a subset of the rows.