Nonparametric Intensity Estimation in Covariate-Driven Poisson Processes Using Deep Learning

We consider the nonparametric estimation problem of the intensity function for a Poisson process with covariates. Because the Kullback-Leibler divergence can diverge in nonparametric estimation, theoretical analysis of the nonparametric maximum likelihood estimator (NPMLE) is challenging. We use a simple unified approach to analyse the NPMLE and derive an oracle inequality in Hellinger distance between the true intensity function and the estimator. As an important application, we derive convergence rates for the NPMLE with deep neural networks. Our analysis shows that the NPMLE with deep learning can mitigate the curse of dimensionality when the true intensity has a composition structure and can automatically adapt to low-dimensional Riemannian manifold structures. Moreover, the derived convergence rates achieve nearly minimax-optimal rates under composition assumptions.

Dec, 2025 12:00 AM