How to model compositional data?
If I understand correctly, in your problem the mineral-element vector $m1$ and $m2$ is known. 

If that is the case, I would suggest LASSO regression. Suppose you have $n$ minerals $m_1$, $m_2$, ..., $m_n\in\mathbb{R}^p$ arranged into a matrix $M=[m_1;m_2,...,m_n]\in\mathbb{R}^{n\times p}$, and an unknown mixing vector $x\in\mathbb{R}^n$ (in your case, $x=[0.25; 0.75]$). Your sensor measures $y=M^\top x\in\mathbb{R}^n$ and you want to recover $x$ from $y$.

The LASSO regression aims to solve $\hat{x}=\arg\min_x ||y-M^\top x||_2 + \lambda||x||_1$. The $\ell_1$ penalty promotes sparsity in $x$ and could be beneficial when you have a lot of candidate minerals.

I don't think you need any special handling due to the compositional nature of the data, other than normalizing the final $\hat{x}$ if needed.