Multivariate Hawkes processes are a widely used class of self-exciting point processes, but maximum likelihood estimation naively scales as $O(N^2)$ in the number of events. The canonical linear exponential Hawkes process admits a faster $O(N)$ recurrence, but prior work evaluates this recurrence sequentially, without exploiting parallelization on modern GPUs. We show that the Hawkes process intensity can be expressed as a product of sparse transition matrices admitting a linear-time associative multiply, enabling computation via a parallel prefix scan. This yields a simple yet massively parallelizable algorithm for maximum likelihood estimation of linear exponential Hawkes processes. Our method reduces the computational complexity to approximately $O(N/P)$ with $P$ parallel processors, and naturally yields a batching scheme to maintain constant memory usage, avoiding GPU memory constraints. Importantly, it computes the exact likelihood without any additional assumptions or approximations, preserving the simplicity and interpretability of the model. We demonstrate orders-of-magnitude speedups on simulated and real datasets, scaling to thousands of nodes and tens of millions of events, substantially beyond scales reported in prior work. We provide an open-source PyTorch library implementing our optimizations.

How to cluster event sequences generated via different point processes is an interesting and important problem in statistical machine learning. To solve this problem, we propose and discuss an effective model-based clustering method based on a novel Dirichlet mixture model of a special but significant type of point processes --- Hawkes process. The proposed model generates the event sequences with different clusters from the Hawkes processes with different parameters, and uses a Dirichlet process as the prior distribution of the clusters. We prove the identifiability of our mixture model and propose an effective variational Bayesian inference algorithm to learn our model. An adaptive inner iteration allocation strategy is designed to accelerate the convergence of our algorithm. Moreover, we investigate the sample complexity and the computational complexity of our learning algorithm in depth. Experiments on both synthetic and real-world data show that the clustering method based on our model can learn structural triggering patterns hidden in asynchronous event sequences robustly and achieve superior performance on clustering purity and consistency compared to existing methods.

Additionally, we conduct ablation studies to explore the robustness of our method concerning various hyperparameters.

This paper discusses one of the most fundamental issues about point processes that what is the best sampling method for point processes. We propose thinning as a downsampling method for accelerating the learning of point processes.

There are plenty of previous studies targeting the problem from different aspects. For temporal point process, agreat number of works [3, 13, 15, 16, 28] attempt to model the intensify function from statistic views, and recent studies harness deep recurrent model [24], generative adversarial network [23] and reinforcement learning [19, 18] to learn the temporal process. These researches mainly focus on one-dimension eventsequences where eacheventpossesses thesame marker.

Real-world event sequences consist of complex mixtures of different types of events occurring intime. Aneventmay depend onpast events ofthesame type, as well as, the other types.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Indeed, the results in Table 1, which shows13 the mean absolute percentage errors (MAPE), demonstrates this. The ac-14 curacy of neural ODE for the Poisson process is on par with our neural15 JSDE. However, for the Hawkes process (Exponential), Hawkes process16 (Power-Law), and self-correcting process, neural ODE gives much larger17 predictions errors. Forthesocial/medicaldatasets,weuseda20/64-24 dimensional latent state and parameterized the functions with two-hidden-layer MLPs with 32/64 hidden units. The time series modeling software that we used is designed for long event sequences and ignores the idle time after31 thelastevent.