I was reading a paper, and came across this argument: 
H and W are matrices here, and $\Pi_k$ is a scalar. I am not able to understand what makes the rank of A arbitrarily high rank? I don't follow the logic that log_sum_exp is a non-linear function so $\hat{A}_{MoS}$ is arbitrarily high rank.
A is basically a sum of matrices here, each of those matrices have a bounded rank. How come A has infinite rank then?
The equation is taken from this paper: https://arxiv.org/abs/1711.03953 (sec 2.4)