Can somebody please tell me if there is any iterative algorithm to solve the following optimization problem:
Given a collection of vectors in a matrix $X =\left\{x_i\right\}_{i=1}^q \in\mathbf{C}^{n\times q} $ such that $n>q$ and some other matrix $B \in \mathbf{C}^{m\times n}$ such that $m >n$, how do I solve for $S \in \mathcal{S}$ such that the norm $\vert \vert B^H S BX\vert \vert_{??}$ is minimized subject to the following condition:
$y_i:= B^H S Bx_i$ is as sparse as possible.
$ B^H SB$ is unitary
$\mathcal{S} = \left\{ S\in \mathbb{R}^{m \times m} \mid S \quad \text{is diagonal} \right\}$
I would appreciate it if somebody could suggest me the norm that I could use for minimization and also the restriction on the set $\mathcal{S}$.