利用神经网络插件分类扩散过程中的漂移函数
Plug-In Classification of Drift Functions in Diffusion Processes Using Neural Networks
AI总结：
本文研究扩散过程的监督多类分类，通过神经网络估计类特定漂移函数构建插件分类器，分析了漂移估计、时间离散化和维度对分类风险的影响，并通过数值实验验证了方法的有效性。
摘要：
我们研究了扩散过程的监督多类分类，其中每个类别由不同的漂移函数特征化，轨迹在离散时间点观测。我们首先推导了多维贝叶斯规则，然后通过神经网络估计类特定的漂移函数来构建插件分类器。在标准正则性假设下，我们建立了分类风险的收敛率，明确指出了漂移估计、时间离散化和维度的贡献。我们的分析还突显了利用扩散结构的优势：漂移从所有观测增量中学习，导致比直接轨迹基于神经分类器在所考虑设置中更严格的保证。数值实验支持理论：所提出的方法在一维情况下优于Denis等人（2024）的方法，在更高维度时当漂移函数具有组合结构时仍有效，并优于直接在轨迹上训练的端到端神经分类器，如Bos & Schmidt-Hieber（2022）的方法。
英文摘要：
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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