加权弗罗贝尼乌斯损失能有效抑制低频误差，提升迭代求解器性能。
Spectral Analysis of the Weighted Frobenius Objective
- 采用加权弗罗贝尼乌斯损失，强化对小特征值方向误差的惩罚。
- 在固定误差预算下，最优解仅存在于最大特征值方向。
- 适用于多种分解方式，无需神经网络，可直接训练稀疏因子。
本文研究了在预条件迭代求解器背景下，用于逼近对称正定矩阵的加权弗罗贝尼乌斯损失。与标准弗罗贝尼乌斯范数不同，该损失对系统矩阵小特征值对应的误差分量施加更强惩罚。分析表明，每个特征模态被其对应特征值的平方缩放，且在固定误差预算下，损失仅在误差局限于最大特征值方向时最小化。这一结果为最小化加权损失能自然抑制低频分量提供了严格解释，对共轭梯度法具有实际意义。该分析不依赖具体近似方法或稀疏模式，适用于不完全分解、代数更新和基于学习的构造。数值实验验证了理论预测，包括通过直接梯度更新训练稀疏因子至IC(0)形式，未使用任何神经网络模型。
原文摘要 · Abstract (English)
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.
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