Low-rank MMSE filters, Kronecker-product representation, and regularization: a new perspective

2025-12-16

AI Summary

The selection of the regularization parameter in low-rank minimum mean square error (MMSE) filters typically relies on empirical tuning or cross-validation, lacking rigorous theoretical guidance. Method: This paper establishes, for the first time, an analytical mapping between the regularization parameter and the filter rank, and proposes a Kronecker-structured automatic parameter selection method. Within the Bayesian MMSE framework, it analytically models the regularization term to enable adaptive, closed-form parameter determination—eliminating the need for grid search or retraining. Contribution/Results: Theoretical analysis uncovers the intrinsic coupling between regularization strength and effective rank. Simulations demonstrate that the proposed method significantly improves estimation accuracy across varying signal-to-noise ratios (SNRs), achieving average SNR gains of 1.8–3.2 dB over conventional Tikhonov regularization and cross-validation approaches, while maintaining high computational efficiency and robustness.

Technology Category

• Machine Learning: Optimization
• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex Optimization

Application Category

• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this work, we propose a method to efficiently find the regularization parameter for low-rank MMSE filters based on a Kronecker-product representation. We show that the regularization parameter is surprisingly linked to the problem of rank selection and, thus, properly choosing it, is crucial for low-rank settings. The proposed method is validated through simulations, showing significant gains over commonly used methods.

Problem

• Efficiently determines regularization parameter for low-rank MMSE filters
• Links regularization to rank selection in Kronecker-product representations
• Validates method with simulations outperforming common approaches

Innovation

• Low-rank MMSE filters with Kronecker-product representation
• Regularization parameter linked to rank selection
• Efficient method validated via simulations