Exact, Fast and Expressive Poisson Point Processes via Squared Neural
  Families
Russell Tsuchida Cheng Soon Ong Dino Sejdinovic
We introduce squared neural Poisson point processes (SNEPPPs) by
parameterising the intensity function by the squared norm of a two layer neural
network. When the hidden layer is fixed and the second layer has a single
neuron, our approach resembles previous uses of squared Gaussian process or
kernel methods, but allowing the hidden layer to be learnt allows for
additional flexibility. In many cases of interest, the integrated intensity
function admits a closed form and can be computed in quadratic time in the
number of hidden neurons. We enumerate a far more extensive number of such
cases than has previously been discussed. Our approach is more memory and time
efficient than naive implementations of squared or exponentiated kernel methods
or Gaussian processes. Maximum likelihood and maximum a posteriori estimates in
a reparameterisation of the final layer of the intensity function can be
obtained by solving a (strongly) convex optimisation problem using projected
gradient descent. We demonstrate SNEPPPs on real, and synthetic benchmarks, and
provide a software implementation. https://github.com/RussellTsuchida/snefy