Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization

2026-05-18

AI Summary

This work addresses the challenge of inferring dynamic graph structures from high-dimensional time-series observations by simultaneously preserving temporal continuity and respecting the geometric constraints imposed by the Riemannian manifold of precision matrices. The authors propose DEGFM, a novel algorithm that uniquely integrates an elliptical graphical factor model—characterized by a low-rank plus diagonal precision matrix—with geodesic regularization on the Grassmann manifold. This approach enables stable estimation under limited sample sizes and solves the resulting non-convex optimization problem via Riemannian gradient descent, rigorously maintaining manifold consistency. Extensive experiments on both synthetic and real-world datasets demonstrate that DEGFM significantly outperforms existing methods across multiple evaluation metrics, confirming its effectiveness and robustness.

Abstract

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.

Problem

• time-varying graph
• temporal coherence
• Riemannian geometry
• precision matrix
• symmetric positive definite manifold

Innovation

• Riemannian optimization
• Grassmann manifold
• time-varying graph
• low-rank-plus-diagonal
• geodesic regularization