ARFs are the basis for two further algorithms, FORests for
Density Estimation (FORDE) and FORests for GEnerative
modeling (FORGE). We present pseudocode for both (see
Algs. 2 and 3). The key point to recognize is that, un-
der the local independence criterion, joint densities can be
learned by running d separate univariate estimators within
each leaf. This is exponentially easier than multivariate den-
sity estimation, which suffers from the notorious curse of
dimensionality. Summarizing the challenges with estimat-
ing joint densities, one recent textbook on KDE concludes
that “nonparametric methods for kernel density problems
should not be used for high-dimensional data and it seems
that a feasible dimensionality should not exceed ﬁve or six...”
(Gramacki, 2018, p. 60). By contrast, our method scales
much better with data dimensionality, exploiting the ﬂexi-
bility of ARFs to learn an independence-inducing partition
that renders density estimation relatively straightforward.
Adversarial Random Forests for Density Estimation and Generative Modeling