Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Michael L. Wells, Kamel Lahouel, Bruno Jedynak

TL;DR

The paper develops SOCK, a kernel-based two-step approach for learning multivariate SDEs by first estimating the drift in an RKHS via vector-valued occupation kernels and then estimating the diffusion through operator-valued occupation kernels constrained to be PSD. It presents implicit and explicit kernel formulations, each stabilized by Fenchel duality (or projections) to avoid intractable likelihoods, and proves representer theorems that yield finite-dimensional optimizations. Through extensive experiments on simulated dynamics, brain-amyloid data, and stochastic SIR trajectories, SOCK shows competitive drift estimates and superior diffusion-estimation accuracy relative to several baselines, indicating strong predictive and descriptive power in noisy, high-dimensional settings. The framework provides scalable strategies (Nyström, inducing points, explicit kernels) and solid theoretical guarantees for the nonparametric recovery of SDE components, with practical applicability to biomedical and complex systems data.

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 subjects.

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

TL;DR

The paper develops SOCK, a kernel-based two-step approach for learning multivariate SDEs by first estimating the drift in an RKHS via vector-valued occupation kernels and then estimating the diffusion through operator-valued occupation kernels constrained to be PSD. It presents implicit and explicit kernel formulations, each stabilized by Fenchel duality (or projections) to avoid intractable likelihoods, and proves representer theorems that yield finite-dimensional optimizations. Through extensive experiments on simulated dynamics, brain-amyloid data, and stochastic SIR trajectories, SOCK shows competitive drift estimates and superior diffusion-estimation accuracy relative to several baselines, indicating strong predictive and descriptive power in noisy, high-dimensional settings. The framework provides scalable strategies (Nyström, inducing points, explicit kernels) and solid theoretical guarantees for the nonparametric recovery of SDE components, with practical applicability to biomedical and complex systems data.

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 subjects.
Paper Structure (26 sections, 16 theorems, 134 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 26 sections, 16 theorems, 134 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup and notation
  3. Method
  4. Estimating the drift
  5. Estimating the diffusion
  6. Implicit kernel
  7. Explicit kernel
  8. Computational complexity
  9. Drift
  10. Diffusion (implicit)
  11. Diffusion (explicit)
  12. Techniques to reduce the complexity for large datasets
  13. Experiments
  14. Datasets
  15. Hyperparameter tuning
  16. ...and 11 more sections

Key Result

Theorem 3.1

Suppose that the following regularity condition holds: for $i=0,\ldots, n-1$ where $\hbox{\rm Tr}(\cdot)$ denotes the matrix trace. Then there exists a minimizer $f^* \in H$ of eq:drift_cost given as a linear combination of functions $L_i^*$: where $\alpha_i \in \mathbb{R}^d$ and $L_i^* \colon \mathbb{R}^d \to \mathbb{R}^{d \times d}$ is the function defined by for $i=0,\ldots, n-1$. The coeffi

Figures (3)

  • Figure 2: Dense matrix-valued diffusion dataset. Plot of train and validation sets. Training data is in blue and validation data is in is red.
  • Figure 3: Dense matrix-valued diffusion dataset. (\ref{['fig:drift_est']}) Plot of estimated drift in black and true drift in grey. Training data is in the background. (\ref{['fig:diff_est']}) Plot of estimated $a$ in black and true function $\sigma_0\sigma_0^T$ in grey. The matrix-valued output is represented as ellipses. Training data is in the background.
  • Figure 4: Exponential dynamics dataset. (\ref{['fig:drift_est_1D']}) Plot of estimated drift in blue and true drift in black. (\ref{['fig:diff_est_1D']}) Plot of estimated $a$ in blue and true function $\sigma_0^2$ in black.

Theorems & Definitions (27)

  • Theorem 3.1: Representer theorem
  • Proposition 3.2
  • Theorem 3.3: Representer theorem
  • Theorem 3.4
  • Theorem 3.5
  • Proposition A.1
  • Proof 1
  • Theorem B.1
  • Proof 2
  • Proposition B.2
  • ...and 17 more