Column Subset Selection Problem is UG-hard

We address two problems related to selecting an optimal subset of columns from a matrix. In one of these problems, we are given a matrix A∈Rm×n and a positive integer k, and we want to select a sub-matrix C of k columns to minimize ‖A−ΠCA‖F, where ΠC=CC+ denotes the matrix of projection onto the space spanned by C. In the other problem, we are given A∈Rm×n, positive integers c and r, and we want to select sub-matrices C and R of c columns and r rows of A, respectively, to minimize ‖A−CUR‖F, where U∈Rc×r is the pseudo-inverse of the intersection between C and R. Although there is a plethora of algorithmic results, the complexity of these problems has not been investigated thus far. We show that these two problems are NP-hard assuming UGC.