The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations

2025-05-16

AI Summary

This work addresses model-free learning for multivariate stochastic differential equations (SDEs), focusing on nonparametric estimation of the unknown drift function and diffusion matrix. We propose a two-stage kernel-based framework: first, a vector-valued occupation kernel is employed in a reproducing kernel Hilbert space (RKHS) to estimate the drift term; second, we introduce a novel operator-valued occupation kernel to model the diffusion function as a positive semidefinite operator, enabling direct structural learning. To circumvent the intractable likelihood of SDEs, we formulate a reconstruction-error-based objective and leverage Fenchel duality for efficient optimization. The method combines theoretical rigor with computational scalability. Empirical evaluation on synthetic benchmarks and real-world Alzheimer’s disease amyloid PET imaging data demonstrates high predictive accuracy and strong generalization performance.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We present a novel kernel-based method for learning multivariate stochastic differential equations (SDEs). The method follows a two-step procedure: we first estimate the drift term function, then the (matrix-valued) diffusion function given the drift. Occupation kernels are integral functionals on a reproducing kernel Hilbert space (RKHS) that aggregate information over a trajectory. Our approach leverages vector-valued occupation kernels for estimating the drift component of the stochastic process. For diffusion estimation, we extend this framework by introducing operator-valued occupation kernels, enabling the estimation of an auxiliary matrix-valued function as a positive semi-definite operator, from which we readily derive the diffusion estimate. This enables us to avoid common challenges in SDE learning, such as intractable likelihoods, by optimizing a reconstruction-error-based objective. We propose a simple learning procedure that retains strong predictive accuracy while using Fenchel duality to promote efficiency. We validate the method on simulated benchmarks and a real-world dataset of Amyloid imaging in healthy and Alzheimer's disease (AD) subjects.

Problem

• Learning multivariate stochastic differential equations (SDEs) using kernel-based methods
• Estimating drift and diffusion functions in SDEs without intractable likelihoods
• Validating method on simulated benchmarks and Alzheimer's disease data

Innovation

• Kernel-based method for learning SDEs
• Vector-valued occupation kernels for drift estimation
• Operator-valued kernels for diffusion estimation