🤖 AI Summary
Current spectral learning methods in scientific machine learning yield only point estimates, lacking uncertainty quantification—thus failing to meet trustworthiness requirements in safety-critical applications. To address this, we propose the Bayesian Parameterized Matrix Model (B-PMM), the first framework enabling Bayesian spectral uncertainty modeling under Hermitian constraints. Methodologically, B-PMM integrates adaptive spectral decomposition, regularization-aware matrix perturbation bounds, and manifold-aware Gaussian variational inference to ensure both geometric consistency and statistical reliability. Theoretically, we establish finite-sample calibration guarantees dependent on the spectral gap. Empirically, B-PMM achieves an expected calibration error (ECE) < 0.05 across matrices ranging from 5×5 to 500×500, demonstrating superior calibration and robust degradation behavior under spectral ill-conditioning.
📝 Abstract
Scientific machine learning increasingly uses spectral methods to understand physical systems. Current spectral learning approaches provide only point estimates without uncertainty quantification, limiting their use in safety-critical applications where prediction confidence is essential. Parametric matrix models have emerged as powerful tools for scientific machine learning, achieving exceptional performance by learning governing equations. However, their deterministic nature limits deployment in uncertainty quantification applications. We introduce Bayesian parametric matrix models (B-PMMs), a principled framework that extends PMMs to provide uncertainty estimates while preserving their spectral structure and computational efficiency. B-PMM addresses the fundamental challenge of quantifying uncertainty in matrix eigenvalue problems where standard Bayesian methods fail due to the geometric constraints of spectral decomposition. The theoretical contributions include: (i) adaptive spectral decomposition with regularized matrix perturbation bounds that characterize eigenvalue uncertainty propagation, (ii) structured variational inference algorithms using manifold-aware matrix-variate Gaussian posteriors that respect Hermitian constraints, and (iii) finite-sample calibration guarantees with explicit dependence on spectral gaps and problem conditioning. Experimental validation across matrix dimensions from 5x5 to 500x500 with perfect convergence rates demonstrates that B-PMMs achieve exceptional uncertainty calibration (ECE < 0.05) while maintaining favorable scaling. The framework exhibits graceful degradation under spectral ill-conditioning and provides reliable uncertainty estimates even in near-degenerate regimes. The proposed framework supports robust spectral learning in uncertainty-critical domains and lays the groundwork for broader Bayesian spectral machine learning.