  Decision forests (Forests), in particular random forests and gradient
boosting trees, have demonstrated state-of-the-art accuracy compared to other
methods in many supervised learning scenarios. In particular, Forests dominate
other methods in tabular data, that is, when the feature space is unstructured,
so that the signal is invariant to a permutation of the feature indices.
However, in structured data lying on a manifold (such as images, text, and
speech) deep networks (Networks), specifically convolutional deep networks
(ConvNets), tend to outperform Forests. We conjecture that at least part of the
reason for this is that the input to Networks is not simply the feature
magnitudes, but also their indices. In contrast, naive Forest implementations
fail to explicitly consider feature indices. A recently proposed Forest
approach demonstrates that Forests, for each node, implicitly sample a random
matrix from some specific distribution. These Forests, like some classes of
Networks, learn by partitioning the feature space into convex polytopes
corresponding to linear functions. We build on that approach and show that one
can choose distributions in a manifold-aware fashion to incorporate feature
locality. We demonstrate the empirical performance on data whose features live
on three different manifolds: a torus, images, and time-series. Moreover, we
demonstrate its strength in multivariate simulated settings and also show
superiority in predicting surgical outcome in epilepsy patients and predicting
movement direction from raw stereotactic EEG data from non-motor brain regions.
In all simulations and real data, Manifold Oblique Random Forest (MORF)
algorithm outperforms approaches that ignore feature space structure and
challenges the performance of ConvNets. Moreover, MORF runs fast and maintains
interpretability and theoretical justification.
