UNIPoint: Universally Approximating Point Processes Intensities

Alexander Soen, Alexander Mathews, Daniel Grixti-Cheng, Lexing Xie

Point processes have been a preferred mathematical tool for describing events over time, and there are many recent approaches for representing and learning them. One notable open question is in precisely describing the flexibility of the various models, and whether there exists a general model that can represent {\em all} point processes. Our work bridges this gap. Focusing on the widely used event intensity function representation of point processes, we provide a constructive 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 positive transfer functions, formulating parameters of piece-wise continuous functions as a dynamic system, and recurrent neural networks 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 datasets show that this simpler representation performs better than Hawkes process variants, and as well as more complex neural network-based approaches. We expect this result will provide a basis for practically selecting and tuning models, as well as further theoretical work on fine-grained characterisation of representational complexity versus expressiveness.

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