Geometry-Aware Langevin Sampling for Matrix-Valued Graph Learning
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910209680605184 |
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| author | Dey, Papri |
| author_facet | Dey, Papri |
| contents | 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 $Φ(W)=-\log\det X(W).$ We show that the Hessian of $Φ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_24913 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometry-Aware Langevin Sampling for Matrix-Valued Graph Learning Dey, Papri Optimization and Control Differential Geometry Dynamical Systems Probability Statistics Theory 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 $Φ(W)=-\log\det X(W).$ We show that the Hessian of $Φ$ 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. |
| title | Geometry-Aware Langevin Sampling for Matrix-Valued Graph Learning |
| topic | Optimization and Control Differential Geometry Dynamical Systems Probability Statistics Theory |
| url | https://arxiv.org/abs/2603.24913 |