Markovian Gaussian Process: A Universal State-Space Representation for Stationary Temporal Gaussian Process

2024-06-29

arXiv.org

AI Summary

Existing methods struggle to efficiently model large-scale temporal neural data and capture dynamic inter-regional brain communication patterns. This paper introduces the Markovian Gaussian Process (Markovian GP), a universal state-space representation that transforms any stationary Gaussian process—single- or multi-output—into a linear dynamical system (LDS), bypassing the restrictive separability assumption on kernels for the first time. Our approach combines spectral density decomposition with discretization of stochastic differential equations, integrating state-space modeling and Bayesian inference. Experiments demonstrate that Markovian GP achieves high accuracy in covariance approximation, regression, and neuroscience applications, while scaling linearly in time complexity—O(N)—and yielding significantly lower error than existing GP-to-LDS conversion methods. By unifying theoretical rigor with computational scalability, it establishes a novel, principled framework for large-scale dynamic brain network modeling.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Gaussian Processes (GPs) and Linear Dynamical Systems (LDSs) are essential time series and dynamic system modeling tools. GPs can handle complex, nonlinear dynamics but are computationally demanding, while LDSs offer efficient computation but lack the expressive power of GPs. To combine their benefits, we introduce a universal method that allows an LDS to mirror stationary temporal GPs. This state-space representation, known as the Markovian Gaussian Process (Markovian GP), leverages the flexibility of kernel functions while maintaining efficient linear computation. Unlike existing GP-LDS conversion methods, which require separability for most multi-output kernels, our approach works universally for single- and multi-output stationary temporal kernels. We evaluate our method by computing covariance, performing regression tasks, and applying it to a neuroscience application, demonstrating that our method provides an accurate state-space representation for stationary temporal GPs.

Problem

• Big Data
• Time-Varying Brain Connectivity
• Neuroscientific Understanding

Innovation

• Adaptive Delay Model
• Markov Gaussian Processes
• Dynamic Brain Communication