I got a following optimization problem
$$\min_{\mathbf{V}}~\frac{1}{2}\|\mathbf{Z}-\mathbf{U}\mathbf{V}\mathbf{X}\|_{F}^{2}+\frac{\lambda}{2}\|\mathbf{V}\|_{F}^{2}$$ where the matrices $\mathbf{Z}\in\mathbb{R}^{N\times T}$, $\mathbf{U}\in\mathbb{R}^{N\times R}$ ($\mathbf{U}^\top\mathbf{U}\neq\mathbf{I}_{R}$), $\mathbf{V}\in\mathbb{R}^{R\times M}$, and $\mathbf{X}\in\mathbb{R}^{M\times T}$ ($\mathbf{X}\mathbf{X}^\top\neq\mathbf{I}_{M}$).
Is there a closed-form solution to the variable $\mathbf{V}$? Thanks for your help in advance!
Note that I know there is a solution by using vectorization and Kronecker product (see Differentiation wrt matrix involvoing Khatri-rao product on Math StackExchange):
$$\hat{\mathbf{V}}=\operatorname{arg}\min_{\mathbf{V}}~\frac{1}{2}\|\operatorname{vec}(\mathbf{Z})-(\mathbf{X}^\top\otimes\mathbf{U})\operatorname{vec}(\mathbf{V})\|_{2}^{2}+\frac{\lambda}{2}\|\operatorname{vec}(\mathbf{V})\|_{2}^{2}$$ where $\otimes$ is the Kronecker product.
But in fact, I want to obtain a solution in the form of matrix.