Exact, Fast and Expressive Poisson Point Processes via Squared Neural Families

TL;DR

This paper introduces squared neural Poisson point processes (SNEPPPs) that parameterize intensity functions with neural networks, offering a flexible, efficient, and closed-form computable approach for modeling point processes, demonstrated on real and synthetic data.

Contribution

The paper presents a novel neural network-based method for Poisson point processes that is more flexible and computationally efficient than previous kernel or Gaussian process methods.

Findings

Closed-form integrated intensity functions in many cases

More memory and time efficient than kernel methods

Effective on real and synthetic benchmarks

Abstract

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