I'd like to show the conditions under which the two are equal?
$$\min_{x_i,y_j}\sum_{ij} |f(x_i,y_j)|^2 \ge \sum_{ij} \min_{x_i,y_j}|f(x_i,y_j)|^2$$
I believe the inequality can be proved as follows: for all $\{x_i,y_j\}$
$$\sum_{ij} |f(x_i,y_j)|^2 \ge \sum_{ij} \min_{x_i,y_j}|f(x_i,y_j)|^2$$
since for each $i,j\in[d]$
$$|f(x_i,y_j)|^2 \ge \min_{x_i,y_j}|f(x_i,y_j)|^2$$