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.

翻译：高斯过程（GPs）与线性动态系统（LDSs）是时间序列与动态系统建模的核心工具。高斯过程能够处理复杂的非线性动态，但计算成本高昂；而线性动态系统虽计算高效，却缺乏高斯过程的表达能力。为融合二者优势，我们提出一种通用方法，使线性动态系统能够精确模拟平稳时序高斯过程。这种状态空间表示称为马尔可夫高斯过程（Markovian GP），它在保持高效线性计算的同时，充分利用了核函数的灵活性。与现有大多数多输出核函数需满足可分离性才能进行GP-LDS转换的方法不同，我们的方法对单输出与多输出平稳时序核函数具有普适性。我们通过协方差计算、回归任务测试及神经科学应用验证了本方法，结果表明该方法能为平稳时序高斯过程提供精确的状态空间表示。