  Though traditional algorithms could be embedded into neural architectures
with the proposed principle of \cite{xiao2017hungarian}, the variables that
only occur in the condition of branch could not be updated as a special case.
To tackle this issue, we multiply the conditioned branches with Dirac symbol
(i.e. $\mathbf{1}_{x>0}$), then approximate Dirac symbol with the continuous
functions (e.g. $1 - e^{-\alpha|x|}$). In this way, the gradients of
condition-specific variables could be worked out in the back-propagation
process, approximately, making a fully functioned neural graph. Within our
novel principle, we propose the neural decision tree \textbf{(NDT)}, which
takes simplified neural networks as decision function in each branch and
employs complex neural networks to generate the output in each leaf. Extensive
experiments verify our theoretical analysis and demonstrate the effectiveness
of our model.
