$W \in \mathbb{R}^{k\times m}$, $H \in \mathbb{R}^{k\times n}$,$Y \in \mathbb{R}^{m\times n}$
$$\min_{W,H}f(W,H) = \|Y-W^T H\|^2_F +\lambda \|W\|^2_F+\lambda \|H\|^2_F$$
$$\frac{\partial f}{\partial W} = 0 \quad \implies \quad W = (H H^T + \lambda I)^{-1} H Y^T$$ $$\frac{\partial f}{\partial H} = 0 \quad \implies \quad H = (W W^T + \lambda I)^{-1} W Y$$
In this figure (matrix factorization with regularization), I try to use the alternating least squares (ALS) algorithm to get the derivative for W when we fix the H as constant and for H when we fix the W as constant, and set the derivative equal to zero, I get the analytical solution for W and H, but it is matrix form.
Can someone help get the vector-form analytical solution? for example: W_i (the ith column of W) and H_i (the ith column of H). Like this: http://bugra.github.io/work/notes/2014-04-19/alternating-least-squares-method-for-collaborative-filtering/
The matrix-format derivatives and vector-format derivatives are same? which one is better?
Hope your help.
Thanks.