In my research of linear algebra and optimization I have recently come across a problem related to the well-known optimization problem:

Given a constant matrix $ X\in\mathbb{R}^{n \times n}$ , $ $ \min_{A\in\mathbb{R}^{n\times k}, B\in\mathbb{R}^{k \times n}} \| X – A B \|_F$ $

Where the norm is the Frobenius norm. I was considering this interesting possible extension:

Given a constant matrix $ X\in\mathbb{R}^{n \times n}$ , $ $ \min_{A\in\mathbb{R}^{n\times k}, B\in\mathbb{R}^{k \times n}} \| X – A \phi(BX) \|_F$ $ where $ \phi $ is a smooth convex function $ \phi : \mathbb{R}\to\mathbb{R} $ (a scalar function applied entrywise to the matrix argument).

Of course if $ \phi $ is linear the problem is quite easy, but when $ \phi$ is non-linear things are more complicated and interesting. My question is, can I hope to find specific functions/families of nonlinear functions $ \phi$ for which the problem is analytically solvable or approximated (by bounds or approximations of some kind)? I was thinking in the direction where $ \phi(x)=\max{(x,0)} $ Also, can someone suggest an algorithm to solve this as CVX threw errors and could not solve it? I thank all helpers and appreciate all assistance