I have a system of equations defined by:
$y^{ij}_m=f(\theta^i_mx^j)$
where $i, j\in \{1, 2\}$, $m\in[1, 2, .. M]$, $y^{ij} \in \mathbb{R}^{M\times 1}$, $\theta^i \in \{0, 1\}^{M\times N}$, $x^j \in \mathbb{R}^{N\times 1}$
$f$ is a sub-linear function.
Also $\overline{\theta^ix^j} = \frac{1}{M}\sum_m \theta^ix^j$ and $\overline{y^{ij}} = \frac{1}{M}\sum_m y^{ij}$
Given: $|\overline{\theta^2x^1}-\overline{\theta^1x^1}| + |\overline{\theta^2x^2}-\overline{\theta^1x^2}| = |\overline{\theta^2x^1}-\overline{\theta^1x^2}| + |\overline{\theta^1x^1}-\overline{\theta^2x^2}|$
I want to find a way to analyse when the following condition will be true: $|\overline{y^{21}}-\overline{y^{11}}| + |\overline{y^{22}}-\overline{y^{12}}| < |\overline{y^{21}}-\overline{y^{12}}| + |\overline{y^{11}}-\overline{y^{22}}|$. More specifically for what properties of $\theta^i$'s and $x^j$'s will this hold true?