I need to solve the following problem
$$\min _{U \in \mathbb{R}^{n \times k}, V \in \mathbb{R}^{k \times d}}\|W \circ(U V-M)\|_{F}^{2}\;\;\;\; \\\text{s.t. }\;\;\;\;\;\;\;\; \text{ trace}(UVG) \leq 0,$$ where $W,M\in\mathbb{R}^{n\times d}$, $G\in \mathbb{R}^{d\times n}$, and $\circ$ denotes the elementwise matrix multiplication.
It is a constrained weighted low-rank approximation problem and I don't really know if it's possible to solve it and what methods to use.
Usually, for the weighted (unconstrained) case an alternating least square approach is used. Maybe a nice idea would be to project the gradients on the feasible set, but I am not sure if that would lead to something good.