Extend the weighted framework to rank regularization

Extend the weighted regularized spectral approximation framework to include rank regularization, which is excluded from the current weak-majorization monotonicity assumptions.

Background

The weighted regularized spectral approximation results assume that the spectral regularizer and constraint are associated with functions monotone under weak majorization on the nonnegative orthant. Rank regularization does not satisfy the required framework and is explicitly identified as an important case lying outside the assumptions.

The authors defer the treatment of rank regularization to future work, making it an explicitly unresolved extension of the weighted low-dimensional reformulation theory developed in the paper.

References

Nevertheless, important cases such as rank regularization lie outside the present framework and are deferred to future work.

— Exact low-dimensional reformulations for regularized spectral approximation  (2608.27052 - Deng et al., 27 Aug 2026) in Section 3.2, subsection “Weighted Regularized Spectral Approximation”