UNIPoint: Universally Approximating Point Processes Intensities

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