稀疏图线性动力系统
Sparse Graphical Linear Dynamical Systems
AI总结：
针对线性高斯SSM缺乏静态与动态图联合建模的问题，提出融合graphical Lasso与因果图的框架及DGLASSO推断算法，并验证其收敛性和有效性。
摘要：
时间序列数据集在机器学习中处于核心地位，并应用于生物医学、地球观测和网络分析等众多科学与工程领域。关于状态空间模型（SSM）已有大量研究，这类模型是强大的数学工具，能够对时间序列进行概率化且可解释的学习。学习SSM中的模型参数可以说是最复杂的任务之一，而已知纳入先验知识既有助于解释，也会使推断任务复杂化。近期的研究尝试将图视角引入其中一些模型参数，但它们存在显著局限，而本工作解决了这些局限。更一般地说，现有图建模工具要么被设计为纳入静态信息，关注独立随机变量之间的统计依赖（例如graphical Lasso方法）；要么被设计为纳入动态信息，强调时间序列样本之间的因果关系（例如graphical Granger方法）。然而，在SSM背景下，尚不存在将静态与动态图建模结合起来的联合方法。本工作提出一种新颖方法来填补这一空白：通过引入一个联合图建模框架，将graphical Lasso模型与用于线性高斯SSM的基于因果的图方法连接起来。我们提出DGLASSO（Dynamic Graphical Lasso，动态图Lasso），这是该框架内的一种新推断方法，实现了高效的分块交替优化-最小化（majorization-minimization）算法。该算法借助非线性分析中的现代工具并在此基础上加以发展，从而建立了收敛性。在各种合成数据上的实验验证展示了所提模型和推断算法的有效性。
英文摘要：
Time-series datasets are central in machine learning with applications in numerous fields of science and engineering, such as biomedicine, Earth observation, and network analysis. Extensive research exists on state-space models (SSMs), which are powerful mathematical tools that allow for probabilistic and interpretable learning on time series. Learning the model parameters in SSMs is arguably one of the most complicated tasks, and the inclusion of prior knowledge is known to both ease the interpretation but also to complicate the inferential tasks. Very recent works have attempted to incorporate a graphical perspective on some of those model parameters, but they present notable limitations that this work addresses. More generally, existing graphical modeling tools are designed to incorporate either static information, focusing on statistical dependencies among independent random variables (e.g., graphical Lasso approach), or dynamic information, emphasizing causal relationships among time series samples (e.g., graphical Granger approaches). However, there are no joint approaches combining static and dynamic graphical modeling within the context of SSMs. This work proposes a novel approach to fill this gap by introducing a joint graphical modeling framework that bridges the graphical Lasso model and a causal-based graphical approach for the linear-Gaussian SSM. We present DGLASSO (Dynamic Graphical Lasso), a new inference method within this framework that implements an efficient block alternating majorization-minimization algorithm. The algorithm's convergence is established by departing from modern tools from nonlinear analysis. Experimental validation on various synthetic data showcases the effectiveness of the proposed model and inference algorithm.
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