Efficient Temporal Point Processes via Monotone Alternating Splines

Temporal point processes (TPPs) are essential for modeling event sequences in continuous time. While modeling the Cumulative Conditional Intensity Function (CCIF) is computationally more efficient than modeling the conditional intensity function (CIF) directly, current CCIF parameterizations rely on Monotone Neural Networks (MNNs). This paper investigates the structural deadlocks of MNNs—specifically convexity restrictions, saturation limits, and mathematical violations—that hinder their ability to represent complex temporal dynamics.

The authors propose Monotone Alternating Splines (MAS), a framework that replaces rigid MNNs with piecewise monotone splines. MAS partitions the CCIF into two components: an interpolation component that uses fine-grained monotone splines to fit local dynamics with C1-continuity, and an extrapolation component that ensures global monotonicity over an unbounded time horizon. By using analytical derivatives from the splines, MAS eliminates the need for numerical integration or automatic differentiation during likelihood evaluation, significantly improving computational efficiency.

MAS addresses the three structural deadlocks of MNNs by providing a flexible, piecewise-defined architecture that does not force the intensity function into a specific shape (e.g., strictly convex or decaying). Theoretical analysis establishes that MAS possesses superior approximation capabilities compared to MNNs, with generalization error bounds that explicitly decompose the contributions of interpolation, extrapolation, and model complexity. Extensive experiments on both synthetic and real-world datasets show that MAS consistently outperforms state-of-the-art TPP models in negative log-likelihood, timestamp prediction accuracy, and computational training speed.

This work provides a robust, mathematically grounded alternative to MNNs for TPP modeling. By resolving the fundamental expressivity bottlenecks of previous CCIF-based methods, MAS enables more accurate forecasting of event sequences in domains ranging from social networks to finance and seismology, while simultaneously reducing the computational burden of training deep TPP models.