Geometry-Aware Langevin Sampling for Matrix-Valued Graph Learning

Bayesian inference over positive semidefinite (PSD) matrix-valued parameters arises in structured covariance estimation, graph-Laplacian precision models, and multi-output graph learning, but Euclidean proposals often mix poorly near the cone boundary. We propose ConeMALA, a geometry-aware Metropolis-adjusted Langevin algorithm whose proposal geometry is induced by the model's log-determinant structure. For a PSD-weighted graph with edge kernels Wesucceq 0, block Laplacian L(W) , and stabilizer Rsucc 0, the lifted precision matrix X(W)=L(W)+R∈ mathbb S++md defines the log-determinant energy Phi(W)=-logdet X(W). We show that the Hessian of Phi is the pullback of the affine-invariant SPD metric under the map Wmapsto X(W), yielding explicit intrinsic Langevin proposals with Metropolis-Hastings correction using the closed-form SPD exponential-map Jacobian. We validate the metric on rank-one PSD edge perturbations for d=5, obtaining essentially exact agreement between analytic curvature scores and finite-difference curvatures. In intrinsic SPD posterior and matrix-valued graph Gaussian experiments, ConeMALA achieves stable multichain diagnostics and substantially higher ESS/sec than Euclidean MALA and generic RMALA, while a PDHMC-like finite-difference baseline is accurate but computationally prohibitive at larger graph sizes. These results show that pullback log-determinant geometry provides a practical route to uncertainty quantification in PSD-constrained graph learning.