AI 中文总结
该研究针对正则化谱近似问题及其加权变体，利用弱控制单调性、相容性与同时对角化条件建立了精确低维重构，并通过反例验证了同时对角化条件的必要性。
AI 中文摘要
我们研究正则化谱近似问题及其加权变体的精确低维重构。在无加权情形下，保真项的弱控制单调性使我们能在奇异值上实现精确重构；在加权情形下，我们在相容性与同时对角化条件下建立了该重构，还通过反例证明，若缺少同时对角化，该精确重构可能失效。
英文摘要
We study exact low-dimensional reformulations for a regularized spectral approximation problem and its weighted variant. In the unweighted setting, weak-majorization monotonicity of the fidelity term enables an exact reformulation over singular values. In the weighted setting, we establish such a reformulation under compatibility and simultaneous diagonalization conditions. We further show, via a counterexample, that this exact reformulation can fail in the absence of simultaneous diagonalization.
