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

TL;DR

This paper introduces a novel Riemannian optimization-based method for estimating dynamic, time-varying graph structures from high-dimensional data, ensuring temporal coherence and respecting the geometry of the underlying manifold.

Contribution

The paper proposes extsc{Degfm}, a new algorithm modeling precision matrices as low-rank-plus-diagonal structures on the Grassmann manifold, with a geodesic penalty for temporal smoothness, and provides convergence guarantees.

Findings

extsc{Degfm} outperforms state-of-the-art methods on synthetic benchmarks.

The method effectively captures smooth graph trajectories respecting Riemannian geometry.

Experiments on real-world data validate the practical utility of the approach.

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.