arXiv:2512.14932v2 Announce Type: replace Abstract: In this work, we propose a method to automatically find the regularization parameter for low-rank MMSE filters based on a Kronecker-product representation, using an approximate leave-one-out (ALO) cross-validation metric. We formulate the problem in a constrained form that covers both error minimization and interference suppression, and explore the link between regularization and rank selection. The proposed method is validated on system identification and antenna beamforming problems, showing significant gains over commonly used methods.