Given the function $$ E(M) = \sum_{i=1}^N \sum_{a=1}^K \left( M_{ia} \cdot \left\lVert\sum_{i=1}^N M_{ia}\cdot x_i\right\rVert_2^2 \right) $$ $x$ is a given constant matrix, $x_i$ is a the $n_\text{th}$ 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.