  Point processes are a useful mathematical tool for describing events over
time, and so there are many recent approaches for representing and learning
them. One notable open question is how to precisely describe the flexibility of
point process models and whether there exists a general model that can
represent all point processes. Our work bridges this gap. Focusing on the
widely used event intensity function representation of point processes, we
provide a proof that a class of learnable functions can universally approximate
any valid intensity function. The proof connects the well known
Stone-Weierstrass Theorem for function approximation, the uniform density of
non-negative continuous functions using a transfer functions, the formulation
of the parameters of a piece-wise continuous functions as a dynamic system, and
a recurrent neural network implementation for capturing the dynamics. Using
these insights, we design and implement UNIPoint, a novel neural point process
model, using recurrent neural networks to parameterise sums of basis function
upon each event. Evaluations on synthetic and real world datasets show that
this simpler representation performs better than Hawkes process variants and
more complex neural network-based approaches. We expect this result will
provide a practical basis for selecting and tuning models, as well as
furthering theoretical work on representational complexity and learnability.
