  Max-stable processes are increasingly widely used for modelling complex
extreme events, but existing fitting methods are computationally demanding,
limiting applications to a few dozen variables. $r$-Pareto processes are
mathematically simpler and have the potential advantage of incorporating all
relevant extreme events, by generalizing the notion of a univariate exceedance.
In this paper we investigate score matching for performing high-dimensional
peaks over threshold inference, focusing on extreme value processes associated
to log-Gaussian random functions and discuss the behaviour of the proposed
estimators for regularly-varying distributions with normalized marginals. Their
performance is assessed on grids with several hundred locations, simulating
from both the true model and from its domain of attraction. We illustrate the
potential and flexibility of our methods by modelling extreme rainfall on a
grid with $3600$ locations, based on risks for exceedances over local quantiles
and for large spatially accumulated rainfall, and briefly discuss diagnostics
of model fit. The differences between the two fitted models highlight the
importance of the choice of risk and its impact on the dependence structure.
