As the title states, I want to get the partial derivative of an NMF optimization problem in order to get the new update rules for $V$ and $V_C$ by using the Karush-KuhnTucker complementary condition.
I want to replace the first term $||X-DV^T||_F^2$ with a new term $||\binom{X}{S^T} - \binom{D}{V_C}V^T||_F^2 $
resulting in an updated optimization problem of:
$$\mathcal{L} = min (D,U,V,V_C, T ≥ 0, p, q) ||\binom{X}{S^T} - \binom{D}{V_C}V^T||_F^2 + \alpha ||Y (A-UTU^T)||_F^2 + \beta tr(H^TLH) + \gamma ||e \circ(\bar BDq-o)||_2^2+ \eta ||D_Lp-y_L||_2^2+ \lambda R$$
where R is ($||D||^2_F + ||V||^2_F + ||V_C||^2_F + ||U||^2_F + ||T||^2_F + ||p||^2_2 + ||q||^2_2$) to avoid overfitting.
I am stuck on getting the partial derivative of $\mathcal{L}$ w.r.t $V$ as well as $\mathcal{L}$ w.r.t $V_C$. and using these respective derivatives to get the new update rules for $V$ and $V_C$. Any help would be appreciated!
Note:
$$X \in \Bbb{R}_+^{n\mathsf{x}t}$$ $$V \in \Bbb{R}_+^{t\mathsf{x}d}$$ $$D \in \Bbb{R}_+^{n\mathsf{x}d}$$ $$S \approx VW^T$$