Joint Topology Learning and Latent Input Identification Using Spatio-Temporally Linear Structured SEM

Topology identification and signal inference are cornerstone tasks in graph signal processing (GSP). Structural Equation Modeling (SEM) is particularly effective for network inference as it explicitly captures causal dependencies. However, a major bottleneck in existing SEM-based approaches is the reliance on fully observable exogenous inputs. In many practical applications, systems are driven by latent stimuli, rendering traditional estimation methods ineffective. To overcome this, we propose a novel SEM framework for the joint inference of graph topology and unknown exogenous inputs. The core innovation lies in the spatio-temporal modeling of these latent inputs: each stimulus is decomposed into a rank-one component characterized by nodal sparsity (spatial localization) and temporal piecewise smoothness (temporal persistence). This structured formulation transforms an otherwise ill-posed blind identification problem into a tractable regularized optimization task. We develop an efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve the resulting convex problem. Numerical experiments on synthetic and real-world datasets demonstrate that the proposed method effectively disentangles endogenous network interactions from latent exogenous influences, outperforming baseline approaches in both topology and signal recovery.