  This study introduces a novel estimation method for the entries and structure
of a matrix $A$ in the linear factor model $\textbf{X} = A\textbf{Z} +
\textbf{E}$. This is applied to an observable vector $\textbf{X} \in
\mathbb{R}^d$ with $\textbf{Z} \in \mathbb{R}^K$, a vector composed of
independently regularly varying random variables, and independent lighter tail
noise $\textbf{E} \in \mathbb{R}^d$. This leads to max-linear models treated in
classical multivariate extreme value theory. The spectral of the limit
distribution is subsequently discrete and completely characterised by the
matrix $A$. Every max-stable random vector with discrete spectral measure can
be written as a max-linear model. Each row of the matrix $A$ is supposed to be
both scaled and sparse. Additionally, the value of $K$ is not known a priori.
The problem of identifying the matrix $A$ from its matrix of pairwise extremal
correlation is addressed. In the presence of pure variables, which are elements
of $\textbf{X}$ linked, through $A$, to a single latent factor, the matrix $A$
can be reconstructed from the extremal correlation matrix. Our proofs of
identifiability are constructive and pave the way for our innovative estimation
for determining the number of factors $K$ and the matrix $A$ from $n$ weakly
dependent observations on $\textbf{X}$. We apply the suggested method to weekly
maxima rainfall and wildfires to illustrate its applicability.
