A common object to describe the extremal dependence of a $d$-variate random vector $X$ is the stable tail dependence function $L$. Various parametric models have emerged, with a popular subclass consisting of those stable tail dependence functions that arise for linear and max-linear factor models with heavy tailed factors. The stable tail dependence function is then parameterized by a $d \times K$ matrix $A$, where $K$ is the number of factors and where $A$ can be interpreted as a factor loading matrix. We study estimation of $L$ under an additional assumption on $A$ called the `pure variable assumption'. Both $K \in \{1, \dots, d\}$ and $A \in [0, \infty)^{d \times K}$ are treated as unknown, which constitutes an unconventional parameter space that does not fit into common estimation frameworks. We suggest two algorithms that allow to estimate $K$ and $A$, and provide finite sample guarantees for both algorithms. Remarkably, the guarantees allow for the case where the dimension $d$ is larger than the sample size $n$. The results are illustrated with numerical experiments and two case studies.