Randomly Masked Linear Autoencoders: A Unified Closed-Form Framework

Randomly masked linear autoencoders (LAEs) reconstruct data from partially observed inputs. Existing formulations, however, do not jointly provide emphasis weighting, independent diagonal control, and an explicit whole-matrix solution. We introduce a unified masked objective that combines emphasis weighting with separately tunable diagonal and full-matrix regularization. Conditioning on whether each target is retained or dropped exposes a shared quadratic structure, yielding a whole-matrix closed form through one shared inverse and diagonal rescaling. Experiments on nine recommendation datasets show that our unified model outperforms existing LAE baselines on most evaluation metrics and achieves competitive accuracy against more complex deep learning methods. Our solver also matches DEQL's column-wise solution numerically at the free-diagonal endpoint while reducing runtime and peak incremental memory. We additionally derive two closed-form low-rank approximations with $O(nk)$ factorized storage. Our code is available at https://anonymous.4open.science/r/ICLR_2027_Unified_Recommendation-E5F2.