The main objective function for my non-negative matrix factorization (NMF) algorithm is Kullback-Leibler divergence where the approximation of $V$ by $V \simeq WH$ is achieved by minimizing the error function $D(V ||WH)$.
$D(V ||WH)$ can be derived by majoriztion-minimization:
$D(V ||WH) \leq -\sum_{ij} V_{ij} \sum_{k} \pi_{ijk} log W_{ik} H_{kj} + \sum_{ij}(WH)_{ij} + const. [1]$
where $V, W, H$ are matrices and the error can be calculated from the difference between $V$ and product of $WH$
And for regularization function for NMF, L1-norm can be majorized as:
$\Lambda \Omega(H_{s}) \leq \Lambda \sum_{g} \langle \frac{\hat{H}}{\epsilon + ||H_{g}||},H_{g} \rangle +const. [2]$
where $\Lambda$ is weight
The question is to measure the error from regularized function $[1] + [2]$, which functions I have to calculate the error from?
Is it the majorized form one: $-\sum_{ij} V_{ij} \sum_{k} \pi_{ijk} log W_{ik} H_{kj} + \sum_{ij}(WH)_{ij} + \Lambda \sum_{g} \langle \frac{\hat{H}}{\epsilon + ||H_{g}||},H_{g} \rangle$ (not $D(V ||WH) + \Lambda \Omega(H_{s})$ itself)? Or is there any techniques for this?