Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships

Inferring time-varying graph structures from high-dimensional nodal observations is a fundamental problem arising in neuroscience, finance, climatology, and beyond. Two intrinsic challenges govern this problem: maintaining the \emph{temporal coherence} of the latent graph across successive observation windows, and respecting the \emph{intrinsic Riemannian geometry} of the symmetric positive definite manifold on which precision matrices naturally reside, a curved space whose geodesic structure departs fundamentally from that of the ambient Euclidean space. In this paper we propose dynamic estimation on the Grassmann manifold with a factor model (\textsc{Degfm}), a novel algorithm that jointly addresses both challenges. We model the time-varying precision matrix sequence as a low-rank-plus-diagonal structure governed by a latent elliptical graph factor model, which drastically reduces the effective parameter count and enables reliable estimation in the challenging small-sample regime. Temporal coherence is enforced through a Riemannian geodesic penalty defined on the Grassmann manifold, ensuring that the estimated graph trajectory is smooth with respect to the intrinsic geometry rather than the ambient Euclidean space. To solve the resulting non-convex optimization problem over Grassmann-manifold-valued sequences subject to the LRaD constraint, we derive an efficient Riemannian gradient descent algorithm that respects the manifold structure at every iterate and rigorously establish its convergence to a stationary point. Extensive experiments on both synthetic benchmarks and real-world datasets demonstrate that \textsc{Degfm} consistently outperforms state-of-the-art baselines across all evaluation metrics, confirming the practical effectiveness of the proposed framework.

Bayesian latent space models offer a principled approach to network representation, but rely on correct specification of both geometry and link function. Real-world networks often violate these assumptions, exhibiting geometric mismatch and structural anomalies that break standard metric properties. We show that such misspecification pushes the data-generating distribution outside the model class, causing Bayesian inference to become overconfident and poorly calibrated. To address this, we propose a generalized posterior framework for random geometric graphs. We introduce Link-Sequential R-SafeBayes, a method that exploits dyadic conditional independence to estimate prequential risk and adaptively tune posterior regularization. Experiments on synthetic and real-world networks demonstrate improved calibration, better link prediction performance, and a reliable criterion for selecting latent geometries across Euclidean, spherical, and hyperbolic spaces.

We propose a novel framework for learning time-varying graphs from spatiotemporal measurements. Given an appropriate prior on the temporal behavior of signals, our proposed method can estimate time-varying graphs from a small number of available measurements. To achieve this, we introduce three regularization terms in convex optimization problems that constrain the sparseness of temporal variations of the time-varying networks. Moreover, a computationally scalable algorithm is introduced to solve the optimization problem efficiently. The experimental results with synthetic and real datasets (point cloud, temperature, and EEG data) demonstrate that our proposed method outperforms state-of-the-art methods.