Smooth Neural Point Processes via B-Splines

Neural temporal point processes (TPPs) are powerful tools for modeling continuous-time event sequences, but they often face a trade-off between model flexibility and computational efficiency. Traditional neural TPPs are frequently constrained by the need to model the compensator (the integral of the intensity function) to avoid numerical integration, which limits architectural choices and forces sequential, slow training. This paper asks whether one can directly model the conditional intensity function (CIF) using B-splines to achieve exact likelihood evaluation while maintaining architectural flexibility and parallel training capabilities.

The researchers represent the CIF as a non-negative linear combination of B-spline basis functions, where the coefficients are predicted by a neural network. This design allows the integral of the CIF (the compensator) to be computed in closed form, eliminating the need for numerical quadrature. Because the B-spline basis functions are non-negative, the model ensures a valid, non-negative intensity function simply by constraining the network's output coefficients. Furthermore, the authors incorporate a roughness penalty based on the integrated squared second derivative of the spline, which helps prevent overfitting and controls local fluctuations in the learned intensity.

The proposed B-spline approach significantly improves computational efficiency, achieving up to a 12x speedup in training time compared to baseline methods by enabling parallel evaluation of event intensities. In experiments across seven synthetic and two real-world datasets, the model demonstrated superior predictive accuracy—measured by the mean absolute error of the next-event time estimator—outperforming existing neural TPP benchmarks in most cases. The results suggest that the combination of B-spline parametrization and smoothness regularization provides a more robust and scalable framework for modeling complex event dynamics.