i was reading the flipout paper and i stumbled over this passage. I will summarize the main point here:
$x_n$ are the inputs, $W$ is the matrix of the weights. such matrix can be decomposed as the sum of matrix of the mean values $\bar W$ and the perturbation matrix $\widehat \Delta W$. $r_n$ and $s_n$ are vectors of +1 and -1
the relation between $y_n$ and $x_n$ is written as
$$y_n =\phi(W^{T}x_n)= \phi((\bar W + \widehat \Delta W \circ r_n s_n^T )^T x_n)= \phi (\bar W^T x_n +(\widehat{\Delta W}^T(x_n \circ s_n))\circ r_n)$$
where $\circ$ denotes element-wise multiplication.
What property has been used in the last passage to transform the equation?
I wanted to report this equation but i am using a different notation ($y_n =\phi(W x_n)$), how does it change?