← 研踪 PaperTrailarXiv:2605.18316 · v2
Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization
首次提交 2026-05-18 20:34
版本更新 2026-11-09 21:12
雷达观测 2026-11-10 15:22
摘要
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 $\min_{k<K}\|\grad F(θ^{(k)})\|^2=\mathcal{O}(1/K)$; (iii) a localized estimator obeys a finite-sample oracle inequality whose error scales with the intrinsic dimension times an effective temporal dimension decreasing from to as coupling strengthens, quantifying the variance reduction and bias of temporal coupling, and extending to misspecified non-LRaD models; (iv) for Student- observations, the local restricted strong convexity is proved rather than assumed, with curvature of order 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.
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v2 · 2026-11-09 21:12 · 最新观测版本
Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization
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 $\min_{k<K}\|\grad F(θ^{(k)})\|^2=\mathcal{O}(1/K)$; (iii) a localized estimator obeys a finite-sample oracle inequality whose error scales with the intrinsic dimension times an effective temporal dimension decreasing from to as coupling strengthens, quantifying the variance reduction and bias of temporal coupling, and extending to misspecified non-LRaD models; (iv) for Student- observations, the local restricted strong convexity is proved rather than assumed, with curvature of order 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.
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