I have an objective function like this:
$\min\limits_G\ f(G) = \rVert A - GG^T \rVert_F^2 = trace(A^TA)-2\ trace(A^TGG^T)+trace(GG^TGG^T)$
For computing $\frac{d f(G)}{d G}$ using [The Matrix Cookbook, Page 13], I know the derivative of the second term is: $-2AG$ because $A$ is a symetric $n \times n$ matrix, but I don't know $\frac{d}{d G} trace(GG^TGG^T)$?
Thanks :)
With $f(G)=\mathrm{Tr}(GG^TGG^T)$ to get the derivative you can compute $f(G+H)-f(G)$ and drop all the terms with more than one $H$ as they are $O(\Vert H\Vert^2)$.
More precisely: $$ f(G+H)-f(G)=\mathrm{Tr}(HG^TGG^T)+\mathrm{Tr}(GH^TGG^T)+\mathrm{Tr}(GG^THG^T)+\mathrm{Tr}(GG^TGH^T)+O(\Vert H\Vert^2)$$ using $\mathrm{Tr}(AB)=\mathrm{Tr}(BA)$ and $\mathrm{Tr}(A^TB)=\mathrm{Tr}(AB^T)$ you can obtain: $$f(G+H)-f(G)=4 \mathrm{Tr}(G^TGG^T H) +O(\Vert H\Vert^2)$$