Plug-In Classification of Drift Functions in Diffusion Processes Using Neural Networks

TL;DR

This work extends diffusion-path classification to multidimensional diffusion processes by introducing a neural-network–based plug-in classifier that learns class-specific drifts from high-frequency trajectory data. Central to the approach is a Bayes classifier characterization using functionals $F_k^*(X)$ and a softmax mapping, which motivates a two-step plug-in construction: discretize the score via $\bar{F}_k$ and estimate the drift functions $b_k$ with sparse neural networks. The authors establish an excess misclassification risk bound that decomposes into a discretization error $\sqrt{\Delta}$ and drift-estimation error, and they derive convergence rates for the NN plug-in under realistic regularity and smoothness assumptions. Numerical experiments in both low and high dimensions show that the NN-based plug-in classifier outperforms B-spline plug-ins and direct end-to-end classifiers, with strong performance even when the drift admits a compositional structure. The results offer a principled, scalable framework for diffusion-structured classification with practical implications for diffusion-based models and time-series analysis.

Abstract

We study a supervised multiclass classification problem for diffusion processes, where each class is characterized by a distinct drift function and trajectories are observed at discrete times. Extending the one-dimensional multiclass framework of Denis et al. (2024) to multidimensional diffusions, we propose a neural network-based plug-in classifier that estimates the drift functions for each class from independent sample paths and assigns labels based on a Bayes-type decision rule. Under standard regularity assumptions, we establish convergence rates for the excess misclassification risk, explicitly capturing the effects of drift estimation error and time discretization. Numerical experiments demonstrate that the proposed method achieves faster convergence and improved classification performance compared to Denis et al. (2024) in the one-dimensional setting, remains effective in higher dimensions when the underlying drift functions admit a compositional structure, and consistently outperforms direct neural network classifiers trained end-to-end on trajectories without exploiting the diffusion model structure.

Figures (3)

• Figure 1: True $b_k, k=1,2,3$ (left) and sample paths of each class (right).
• Figure 2: Classifier comparison: NN-based vs. B-spline-based plug-in (left) and NN-based plug-in vs. direct NN (right).
• Figure 3: Convergence rates of the excess classification risk of NN-based classifier for $d = 2$ (top left), $d = 5$ (top right), $d = 10$ (bottom left), and $d = 50$ (bottom right).

Theorems & Definitions (14)

• Proposition 2.4
• Theorem 2.5
• Theorem 2.7
• Remark 2.8: Compact support assumption
• proof : Proof Sketch of Proposition \ref{['prop:bayes-classifier']}
• proof : Proof Sketch of Theorem \ref{['thm:main-decomposition']}
• Lemma 5.1
• proof : Proof Sketch of Lemma \ref{['lem:relation-E-with-Ek']}
• proof : Proof Sketch of Theorem \ref{['thm:main-NN']}
• proof : Proof of Proposition \ref{['prop:bayes-classifier']}