用拉普拉斯变换降低连续马尔可夫半群学习复杂度
Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups
- 通过转移算子的拉普拉斯变换构造无界生成元的解，避免传统方法局限
- 计算复杂度从二次降至线性，小时间步也能准确学习特征值
- 适用于更广类别的马尔可夫过程，适合高维动态系统建模
马尔可夫过程是众多现实世界随机过程的通用模型。本文提出一种数据驱动方法，通过马尔可夫半群无穷小生成元（IG）的谱分解学习此类模型。由于生成元的无界性，传统方法如向量值回归和希尔伯特-施密特算子分析面临困难。现有技术包括物理信息核回归，计算成本高且适用范围有限，当时间滞后较小时，转移算子方法缺乏恢复保证。我们提出新方法，利用生成元的预解算子，其由转移算子的拉普拉斯变换表征。该方法对时间滞后变化具有鲁棒性，即使在小时间滞后下仍能准确学习特征值。我们的统计分析适用于比现有方法更广泛的马尔可夫过程类别，同时将计算复杂度从状态维度的二次方降至线性。最后，我们在两个实验中展示了该方法的行为表现。
原文摘要 · Abstract (English)
Markov processes serve as a universal model for many real-world random processes. This paper presents a data-driven approach for learning these models through the spectral decomposition of the infinitesimal generator (IG) of the Markov semigroup. The unbounded nature of IGs complicates traditional methods such as vector-valued regression and Hilbert-Schmidt operator analysis. Existing techniques, including physics-informed kernel regression, are computationally expensive and limited in scope, with no recovery guarantees for transfer operator methods when the time-lag is small. We propose a novel method that leverages the IG's resolvent, characterized by the Laplace transform of transfer operators. This approach is robust to time-lag variations, ensuring accurate eigenvalue learning even for small time-lags. Our statistical analysis applies to a broader class of Markov processes than current methods while reducing computational complexity from quadratic to linear in the state dimension. Finally, we illustrate the behaviour of our method in two experiments.
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