Given the function $$ E(M) = \sum_{i=1}^{N}\sum_{a=1}^K(M_{ia}\cdot\lVert\sum_{i=1}^{N}M_{ia}\cdot x_i\rVert_2^2) $$ $x$ is a given constant Matrix, $x_i$ is a the $n_{th}$ column of that. $M_{ia} \in \{0,1\}$ and $\sum_{i=1}^{N}M_{ia}=\frac{N}{K} $, and we also have $\sum_{a=1}^{N}M_{ia}=1 \\\\$.

The question is, can E of M be further simplified to a form using only quadratic and linear terms of M?

Again, thank you so much.