I want to find matrices $X$ such that the diagonal of $X^H X$, where $X^H$ is the Hermitian transpose of $X$, is sparse — as many zeros as possible. Intuitively, $X$ should have many empty columns.
I guess this is one of the sparse matrix recovery problems, but I have trouble describing this constraint in a way that could make it amenable to existing algorithms. Can someone please give me pointers to papers/software?