Closed-form solution for arbitrary LAE regularization weight orders

Determine whether the non-uniformly weighted denoising linear autoencoder objective with diagonal regularization matrix \(\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_k)\) admits a closed-form solution when the weights are in an arbitrary order rather than non-descending order.

Background

The paper discusses a non-uniform weighted $\ell_2$-regularized linear autoencoder objective involving factor matrices $W_1$ and $W_2$, with a diagonal regularization matrix $\Lambda$. Prior work provides a closed-form solution only when the diagonal entries of $\Lambda$ are ordered non-descendingly. The unresolved issue is whether such a closed-form characterization remains available for arbitrary permutations of the regularization weights. Although the paper later claims to resolve this issue through an automatic-sorting result, the passage in A3 explicitly identifies the arbitrary-order case as unresolved in the cited prior work.