I have a question regarding sparse coding, Non-negative sparse coding.
Iterate until convergence:
$ \mathbf{A_i} \leftarrow \arg \! \min_{A \geq 0} || \mathbf{X}_i - \mathbf{B}_i\mathbf{A}||_F^2 + \lambda \sum_{p,q} \mathbf{A}_{pq}$
$\mathbf{B}_i \leftarrow \arg \! \min_{\mathbf{B} \geq 0,||\mathbf{b}^{(j)} ||_2 \leq 1} || \mathbf{X}_i - \mathbf{B} \mathbf{A}_i ||_F^2$
I have a question regarding the algorithm for calculating this value.
- Will I during the operation save all the values for the calculations and then take the one that has the minimum values of what, in total?
- And what does that mean sum all the values?
The paper points to Non-Negative Sparse Coding which solve the problem
$\mathbf{A_i} \leftarrow \arg \! \min_{A \geq 0} ||\mathbf{X}_i - \mathbf{B}_i\mathbf{A}||_2^2 + \lambda ||\mathbf{A}||_{1/2}$
and we get the following algorithm:
$1. \quad \mathbf{B}^0, \mathbf{A}^0, t=0.$
$2. \quad \mathbf{B}' = \mathbf{B}^t - \mu(\mathbf{B}^t \mathbf{A}^t-\mathbf{X})(\mathbf{A}^t)^T.$
$3. \quad \mathbf{A}' = 0.$
$4. \quad \mathbf{B}' = 1, \mathbf{B}^{t+1}=\mathbf{B}'.$
$5. \quad \mathbf{A}^{t+1}=\mathbf{A}^t.*((\mathbf{B}^{t+1})^T\mathbf{A})./((\mathbf{B}^{t+1})^T(\mathbf{B}^{t+1}\mathbf{A}^t+\lambda \mathbf{c})$
$6. \quad t\ +=1$
I do not understand if this algorithm actually solves for A,B or if it just solves the A.
Please help me understand this and please elaborate if you need more information.
I solved it. The algorithm in the paper is a notation for the outcome, nothing else. Seems like alot of papers do this kind of pseudocodes for the mathematics. However the notation is for the paper refered and is also answered by the refered paper.
So in summary, yes the outcome of the algortithm is both A,B.