What is the closed-form solution using ALS algorithm to optimize

[kevin2016]

C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n}

${S}$ and ${P}$ are similarity matrices (symmetric).

$\lambda$, $\alpha$ and $\beta$ are regularized parameters (scalar).

$\circ$ is Hadamard product (element-wise product).

$$\min_{W, H}f(W, H)=\|C\circ(X-WH^T)\|^2_F+\lambda\|W\|^2_F +\lambda\|H\|^2_F + \\\alpha\|S-WW^T\|^2_F +\beta\|P-HH^T\|^2_F$$

For the objective function ${f}$, I used alternating least squares (ALS) algorithm to get the $\frac{\partial f}{\partial W}$ and $\frac{\partial f}{\partial H}$ and set both them to zeros ($\frac{\partial f}{\partial W} = 0$; $\frac{\partial f}{\partial H}=0$), thus I can get the analytical solution for both ${W}$ and ${H}$.

Let set first the $H$ as constant, thus in fact, I will solve such objective function to get $W$.

$$\min_{W, H}f(W, H)=\|C\circ(X-WH^T)\|^2_F+\lambda\|W\|^2_F + \alpha\|S-WW^T\|^2_F$$

Finally, I get

$$\frac{\partial f}{\partial W} = 2[C \circ (WH^T)]H - 2(C \circ X)H +2{\lambda}W + 4{\alpha}WW^TW - 4{\alpha}SW = 0$$, that is

$$[C \circ (WH^T)]H - (C \circ X)H +{\lambda}W + 2{\alpha}WW^TW - 2{\alpha}SW = 0 \quad (1)$$

For equation $(1)$, I can not get the analytical solution for $W$. So can you help me work out:

$$W=?$$

Thanks.

 

[kevin2016]

Hi Greg,

currently, I still can not get the answer about this questions.

In fact, if someone can get the $\frac{\partial f}{\partial W_i}$, wher $W_i$ is the $i$th row of W, it is also OK.

Here is the original paper: "Collaborative matrix factorization with multiple similarities for predictiong drug-target interactions" by xiaodong Zheng and co-workers. In the paper, they give the result (that is $a_i$), but I can not proof their result. Can anybody help?

Thanks.