Let $\mathbb{R}^{m\times n}_r$ be the set of matrices of rank at most $r$ and $\|\cdot\|_f$ be the frobenius norm. Consider two matrices $A_1\in\mathbb{R}^{m_1\times n_1}$ and $A_2\in\mathbb{R}^{m_2\times n_2}$.
Is the following statement true?
$$ \min_{X_1\in \mathbb{R}^{m_1\times n_2}_{r{_1}}\atop X2\in \mathbb{R}^{m_2\times n_2}_{r{_2}}} \left(\|A_1-X_1\|_f+\|A_2-X_2\|_f\right)=\min_{X_1\in \mathbb{R}^{m_1\times n_1}_{r{_1}}} \|A_1-X_1\|_f+\min_{X_2\in \mathbb{R}^{m_2\times n_2}_{r{_2}}} \|A_2-X_2\|_f $$
I'm not sure if I can separate the minimization problem in two.
If anyone has a reference, I would really appreciate it