Empirical risk minimization algorithm for multiclass classification of S.D.E. paths
Christophe Denis, Eddy Ella Mintsa
TL;DR
This work develops a diffusion-path multiclass classifier that treats the drift as class-discriminative and the diffusion as shared but unknown. By formulating an empirical risk minimization problem with a squared (L2) loss and modeling drift and diffusion via B-splines, the authors obtain consistent ERM predictors and derive rates that depend on Hölder smoothness; a margin condition yields fast rates in the binary case. They also introduce an adaptive version to select spline dimensions data-dependently, and demonstrate strong numerical performance on simulated diffusion data, often surpassing plug-in and depth-based baselines. The results advance nonparametric diffusion-path classification and provide practical tools for prediction with diffusion-process features, with potential extensions to time-inhomogeneous and high-dimensional settings.
Abstract
We address the multiclass classification problem for stochastic diffusion paths, assuming that the classes are distinguished by their drift functions, while the diffusion coefficient remains common across all classes. In this setting, we propose a classification algorithm that relies on the minimization of the L 2 risk. We establish rates of convergence for the resulting predictor. Notably, we introduce a margin assumption under which we show that our procedure can achieve fast rates of convergence. Finally, a simulation study highlights the numerical performance of our classification algorithm.