Spectral Analysis of the Weighted Frobenius Objective
This provides a theoretical foundation for improving preconditioning in iterative solvers, but it is incremental as it builds on existing weighted loss concepts.
The paper analyzes a weighted Frobenius loss for approximating symmetric positive definite matrices in preconditioning, showing it penalizes errors in small eigenvalues more strongly and minimizes error in the direction of the largest eigenvalue, with numerical experiments confirming the theory.
We analyze a weighted Frobenius loss for approximating symmetric positive definite matrices in the context of preconditioning iterative solvers. Unlike the standard Frobenius norm, the weighted loss penalizes error components associated with small eigenvalues of the system matrix more strongly. Our analysis reveals that each eigenmode is scaled by the corresponding square of its eigenvalue, and that, under a fixed error budget, the loss is minimized only when the error is confined to the direction of the largest eigenvalue. This provides a rigorous explanation of why minimizing the weighted loss naturally suppresses low-frequency components, which can be a desirable strategy for the conjugate gradient method. The analysis is independent of the specific approximation scheme or sparsity pattern, and applies equally to incomplete factorizations, algebraic updates, and learning-based constructions. Numerical experiments confirm the predictions of the theory, including an illustration where sparse factors are trained by a direct gradient updates to IC(0) factor entries, i.e., no trained neural network model is used.