  We show how neural models can be used to realize piece-wise constant
functions such as decision trees. The proposed architecture, which we call
locally constant networks, builds on ReLU networks that are piece-wise linear
and hence their associated gradients with respect to the inputs are locally
constant. We formally establish the equivalence between the classes of locally
constant networks and decision trees. Moreover, we highlight several
advantageous properties of locally constant networks, including how they
realize decision trees with parameter sharing across branching / leaves.
Indeed, only $M$ neurons suffice to implicitly model an oblique decision tree
with $2^M$ leaf nodes. The neural representation also enables us to adopt many
tools developed for deep networks (e.g., DropConnect (Wan et al., 2013)) while
implicitly training decision trees. We demonstrate that our method outperforms
alternative techniques for training oblique decision trees in the context of
molecular property classification and regression tasks.
