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

원문 초록

Inferring time-varying graphs from high-dimensional nodal observations is challenging when windows contain fewer samples than nodes, observations are heavy-tailed, and the dependence drifts multiplicatively over time. We propose the dynamic elliptical graph factor model (\textsc{Degfm}), combining: an elliptical likelihood that down-weights outliers; a low-rank-plus-diagonal (LRaD) parameterization of each precision matrix, shown to be the posterior precision of a field conditioned on a few aggregate linear measurements; and a temporal regularizer given by the squared affine-invariant distance between consecutive precision matrices, capturing multiplicative evolution and invariant to node rescaling and to the choice between precision and covariance. Estimation on a product of quotient manifolds is solved by a safeguarded Riemannian conjugate-gradient method with closed-form temporal gradients via generalized eigendecompositions. We prove that (i) LRaD is locally identifiable modulo orthogonal transformations under a tangent-space transversality (coherence) condition; (ii) with rank-preserving Armijo backtracking, the solver is well defined and converges to critical points with $\min_{k<K}\|\grad F(\theta^{(k)})\|^2=\mathcal{O}(1/K)$; (iii) a localized estimator obeys a finite-sample oracle inequality whose error scales with the intrinsic dimension $pr+p-r(r-1)/2$ times an effective temporal dimension decreasing from $1$ to $1/T$ as coupling strengthens, quantifying the variance reduction and bias of temporal coupling, and extending to misspecified non-LRaD models; (iv) for Student-$t$ observations, the local restricted strong convexity is proved rather than assumed, with curvature of order $\nu/(p+\nu)$ shown unimprovable; and (v) partial-correlation graphs are recovered exactly by thresholding. Experiments on synthetic data, S\&P~500 returns, and HCP fMRI data validate the approach.