深度神经网络加速泊松过程强度估计的收敛速度
Nonparametric Intensity Estimation for Covariate-Driven Poisson Processes Using Deep Learning
一篇统计学习论文：用 DNN 做泊松过程强度估计，理论证明能接近 minimax 最优速率，还在 Allen Brain 的 Neuropixels 神经放电数据上实测赢了核估计器。
论文研究泊松过程强度函数的非参数最大似然估计（NPMLE），允许强度依赖观测协变量，并避开了 KL 散度风险分析中要求真实强度远离零的限制性假设。作者为协变量平均 Hellinger 距离推导了 oracle 不等式，据此建立基于深度神经网络（DNN）的 NPMLE 收敛速率。当真实强度具有复合结构时，DNN-based NPMLE 达到接近 minimax 最优的收敛速率，无需在网络架构中显式引入该结构，并能自适应事件空间的低维黎曼流形结构。仿真显示 DNN 估计器在多数设置中优于核估计器；在 Allen Brain Observatory Visual Coding Neuropixels 的刺激诱发神经放电数据上，DNN-based NPMLE 的 held-out log-likelihood 优于条件核估计器。
Nonparametric Intensity Estimation for Covariate-Driven Poisson Processes Using Deep Learning
We consider nonparametric maximum likelihood estimation of the intensity function of a Poisson process, allowing the intensity to depend on observed covariates. Such models describe how event rates vary with external conditions, including conditions under which the intensity becomes arbitrarily small or vanishes. However, risk analysis based on the Kullback-Leibler (KL) divergence typically requires the true intensity to be bounded away from zero, a restrictive assumption in applications. We develop a unified approach that avoids this assumption and derive an oracle inequality for the covariate-averaged Hellinger distance between the true intensity and the nonparametric maximum likelihood estimator (NPMLE). As an application of the oracle inequality, we establish convergence rates for NPMLEs based on deep neural networks (DNNs). Our results show that, as in nonparametric regression, the ability of DNNs to capture underlying structure leads to improved convergence rates. Specifically, DNN-based NPMLEs mitigate the curse of dimensionality when the true intensity has a compositional structure and automatically adapt to a low-dimensional Riemannian manifold structure of the event space. For intensity functions with a compositional structure, DNN-based NPMLEs achieve nearly minimax-optimal convergence rates without explicitly incorporating this structure into the network architecture. Simulations demonstrate that DNN-based estimators outperform kernel estimators in most settings, while manifold kernels remain competitive in some designs. Finally, we analyze stimulus-evoked neural spiking using data from the Allen Brain Observatory Visual Coding Neuropixels project. DNN-based NPMLEs provide better out-of-sample performance than a conditional kernel estimator, as evaluated by held-out log-likelihood.