Sparse Graphical Linear Dynamical Systems

Abstract

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.

References in corpus (25)

• Mamba: Linear-Time Sequence Modeling with Selective State Spaces
• KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics
• Latent variable graphical model selection via convex optimization
• Graphical modelling of multivariate time series
• Exact covariance thresholding into connected components for large-scale Graphical Lasso
• Particle Gibbs with Ancestor Sampling
• Semi-Blind Inference of Topologies and Dynamical Processes over Graphs
• A Unified Framework for Structured Graph Learning via Spectral Constraints
• Inference of Spatio-Temporal Functions over Graphs via Multi-Kernel Kriged Kalman Filtering
• A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection
• On efficient adjustment in causal graphs
• Iterated Extended Kalman Smoother-based Variable Splitting for $L_1$-Regularized State Estimation
• Graph Neural Networks in Computer Vision -- Architectures, Datasets and Common Approaches
• A Proximal Approach for a Class of Matrix Optimization Problems
• Graphical Inference in Linear-Gaussian State-Space Models
• Sure Screening for Gaussian Graphical Models
• Sparse Bayesian Estimation of Parameters in Linear-Gaussian State-Space Models
• Learning to Discover Sparse Graphical Models
• Sparse Graph Learning from Spatiotemporal Time Series
• uGLAD: Sparse graph recovery by optimizing deep unrolled networks
• Graph Kalman Filters
• Does the ${\ell }_1$-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?
• The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions
• Implicit Differentiation for Hyperparameter Tuning the Weighted Graphical Lasso
• Differentiable Particle Filters through Conditional Normalizing Flow