用神经网络估计扩散过程的漂移函数，实现高精度分类
Plug-In Classification of Drift Functions in Diffusion Processes Using Neural Networks
- 通过神经网络估计每类的漂移函数，构建可直接使用的分类器
- 在1维和高维场景下均优于已有方法，尤其在组合结构漂移时表现更优
- 适合处理离散观测的扩散过程分类，对数据结构利用更充分
我们研究扩散过程的监督多分类问题，其中每类由不同的漂移函数定义，轨迹在离散时间点被观测。首先推导多维贝叶斯准则，并通过神经网络估计类别特定漂移函数，构造插件分类器。在标准正则性假设下，建立了过剩误分类风险的收敛速率，明确揭示了漂移估计、时间离散化和维度对性能的影响。分析还凸显了利用扩散结构的优势：漂移可从所有观测增量中学习，因此在给定设定下比直接基于轨迹训练的神经分类器获得更紧的保证。数值实验支持理论结果：所提方法在1维情况下优于Denis等（2024）；当漂移函数具有组合结构时，在高维下仍有效；且优于直接在轨迹上训练的端到端神经分类器（如Bos & Schmidt-Hieber, 2022）。
原文摘要 · Abstract (English)
We study supervised multiclass classification for diffusion processes, where each class is characterized by a distinct drift function and trajectories are observed at discrete times. We first derive a multidimensional Bayes rule and then construct a plug-in classifier by estimating the class-specific drifts with neural networks. Under standard regularity assumptions, we establish convergence rates for the excess misclassification risk, making explicit the contributions of drift estimation, time discretization, and dimension. Our analysis also highlights the benefit of exploiting the diffusion structure: the drift is learned from all observed increments, leading to sharper guarantees than direct trajectory-based neural classifiers in the considered setting. Numerical experiments support the theory: the proposed method achieves better classification performance than Denis et al. (2024) in dimension one, remains effective in higher dimensions when the drift functions admit a compositional structure, and outperforms end-to-end neural classifiers trained directly on trajectories, as in Bos & Schmidt-Hieber (2022).
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