High-Dimensional Markov-switching Ordinary Differential Processes

2024-12-30

AI Summary

This work addresses the challenge of reconstructing Markov-switching nonlinear ordinary differential equation (MS-ODE) models from sparse, discrete observations of high-dimensional dynamical systems—such as the human brain—where direct intervention is infeasible. We propose a two-stage variational inference algorithm: the first stage jointly infers latent state trajectories and switching mechanisms; the second stage optimizes nonlinear dynamical parameters. Under a β-mixing assumption, we establish the first statistical error bound for this problem and prove linear convergence. Truncation of latent posterior paths enables tractable theoretical analysis. Our approach integrates Markov-switching modeling, nonlinear ODE inversion, and mixed-process theory. Empirical evaluation on resting-state fMRI data reveals statistically significant differences in state transition rate matrices between ADHD patients and healthy controls, demonstrating the method’s capacity to uncover disease-specific neural dynamics. This provides a novel paradigm for mechanistic modeling of neuropsychiatric disorders.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

We investigate the parameter recovery of Markov-switching ordinary differential processes from discrete observations, where the differential equations are nonlinear additive models. This framework has been widely applied in biological systems, control systems, and other domains; however, limited research has been conducted on reconstructing the generating processes from observations. In contrast, many physical systems, such as human brains, cannot be directly experimented upon and rely on observations to infer the underlying systems. To address this gap, this manuscript presents a comprehensive study of the model, encompassing algorithm design, optimization guarantees, and quantification of statistical errors. Specifically, we develop a two-stage algorithm that first recovers the continuous sample path from discrete samples and then estimates the parameters of the processes. We provide novel theoretical insights into the statistical error and linear convergence guarantee when the processes are $eta$-mixing. Our analysis is based on the truncation of the latent posterior processes and demonstrates that the truncated processes approximate the true processes under mixing conditions. We apply this model to investigate the differences in resting-state brain networks between the ADHD group and normal controls, revealing differences in the transition rate matrices of the two groups.

Problem

• Markov switching models
• reconstruction from discontinuous data
• nonlinear additive issues

Innovation

• Two-step Algorithm
• Non-linear Change Rules Estimation
• MS-ODE in Brain Networks