I am doing a matrix completion problem. Here is my object function:
$$\min_{U\ge0,V\ge0} \frac{\lambda}{2} \Vert \Omega \circ (UV^T - Y) \Vert_F^2 + \frac{1}{2} \Vert U \Vert_F^2 + \frac{1}{2} \Vert V \Vert_F^2$$
I read the post in Derivative of Frobenius norm of Hadamard Product, and I am just confused about the derivative of the first term. I try to derivate the gradient of the first term, here is my derivation:
Let $J(U, V) = \Vert M \Vert_F^2 = \Vert \Omega \circ (UV^T - Y) \Vert_F^2$, then
$$\begin{aligned} dJ &= 2M : dM \\ &= 2M : \Omega \circ d(UV^T - Y) \\ &= 2 \Omega \circ M : d(UV^T-Y) \\ &= 2 \Omega \circ M : (U dV^T + dU V^T) \\ &= 2 (\Omega \circ M)^T U : dV + 2 (\Omega \circ M) V : dU \end{aligned}$$
Let $dV = 0$, then we get
$$\frac{\partial J}{\partial U} = 2 (\Omega \circ M) V$$
Similarly for $V$:
$$\frac{\partial J}{\partial V} = 2 (\Omega \circ M)^T U$$
However, I am confused that $\Omega \circ M = \Omega \circ (\Omega \circ (UV^T - Y)) = \Omega^{\circ 2} \circ (UV^T - Y)$, i.e. I need to element-wise mutiply $\Omega$ twice! Is it correct?
Besides, is it correct to apply "alternating least square" and "multiplicative update rules" here to optimize the cost function? Thank you!