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.

翻译：从高维节点观测中推断时变图结构是神经科学、金融学、气候学等领域的基础性问题。该问题面临两个核心挑战：一是保持连续观测窗口间潜在图的时序一致性，二是尊重精度矩阵自然存在的对称正定流形的内在黎曼几何结构——这一弯曲空间的测地线结构与欧氏空间存在根本差异。本文提出格拉斯曼流形上的因子模型动态估计方法（\textsc{Degfm}），该算法联合解决上述两个挑战。我们采用低秩加对角结构建模时变精度矩阵序列，该结构由潜在椭圆图因子模型控制，显著降低了有效参数数量，使得在困难的小样本场景下仍能实现可靠估计。通过格拉斯曼流形上定义的黎曼测地线罚函数强制执行时序一致性，确保估计图轨迹相对于内在几何结构（而非欧氏空间）保持平滑。为解决受LRaD约束的格拉斯曼流形序列非凸优化问题，我们推导出高效的黎曼梯度下降算法，该算法在每次迭代中尊重流形结构，并严格证明其收敛至稳定点。在合成基准数据集和真实世界数据集上的大量实验表明，\textsc{Degfm}在所有评估指标上均持续超越现有最优基线方法，验证了所提出框架的实际有效性。