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 $W_e\succeq 0$, block Laplacian $L(W)$ , and stabilizer $R\succ 0$, the lifted precision matrix $X(W)=L(W)+R\in \mathbb S_{++}^{md}$ defines the log-determinant energy $\Phi(W)=-\log\det X(W).$ We show that the Hessian of $\Phi$ is the pullback of the affine-invariant SPD metric under the map $W\mapsto X(W)$, 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.