I'm trying to understand the paper The Bipartite Formula Complexity of Inner Product is Quadratic, by Avishay Tal. The argument is recapped here. I am having trouble understanding the proof Theorem 3 from that page.
The approach is to construct an polynomial approximation $p$ for the function $f$, and then show that the matrix $M_f$ is equal to the matrix consisting of sign$(p(x,y))$ for each entry $x,y$.
My question is regarding the following sentence, found on the very last paragraph of the page:
Each summand on the right-hand side corresponds to a matrix of rank 1 over the reals.
I do understand why this must be true. If I construct a matrix with coordinates corresponding to, say, the first summand evaluated at $x,y$ for each row/column $x,y$, I don't see why it must be rank-one.