[Paper Review] Distributionally Robust Receive Combining

shirdel, bizhan|arXiv (Cornell University)|Jan 22, 2024

Direction-of-Arrival Estimation Techniques4 citations

TL;DR

This paper proposes a distributionally robust machine learning framework for wireless signal beamforming and estimation that unifies linear and nonlinear beamformers under a single robust optimization structure, eliminating the need for explicit channel estimation. It proves that ridge and kernel ridge regression are distributionally robust, and demonstrates that diagonal loading and uncertainty-aware estimators (e.g., kernelized diagonal loading) significantly improve performance under limited pilots and multiple uncertainties like noise covariance, channel matrix, and outliers.

ABSTRACT

This article investigates signal estimation in wireless transmission (i.e., receive combining) from the perspective of statistical machine learning, where the transmit signals may be from an integrated sensing and communication system; that is, 1) signals may be not only discrete constellation points but also arbitrary complex values; 2) signals may be spatially correlated. Particular attention is paid to handling various uncertainties such as the uncertainty of the transmit signal covariance, the uncertainty of the channel matrix, the uncertainty of the channel noise covariance, the existence of channel impulse noises, the non-ideality of the power amplifiers, and the limited sample size of pilots. To proceed, a distributionally robust receive combining framework that is insensitive to the above uncertainties is proposed, which reveals that channel estimation is not a necessary operation. For optimal linear estimation, the proposed framework includes several existing combiners as special cases such as diagonal loading and eigenvalue thresholding. For optimal nonlinear estimation, estimators are limited in reproducing kernel Hilbert spaces and neural network function spaces, and corresponding uncertainty-aware solutions (e.g., kernelized diagonal loading) are derived. In addition, we prove that the ridge and kernel ridge regression methods in machine learning are distributionally robust against diagonal perturbation in feature covariance.

Motivation & Objective

• To unify existing linear and nonlinear beamformers under a single robust optimization framework that handles multiple uncertainties in wireless signal estimation.
• To eliminate the need for explicit channel estimation by formulating beamforming and estimation as a distributionally robust learning problem.
• To generalize classical methods like diagonal loading and eigenvalue thresholding into a distributionally robust setting with theoretical justification.
• To extend robustness theory to nonlinear estimators by constraining them to reproducing kernel Hilbert spaces and neural network function spaces.
• To provide a unified framework that handles signal covariance uncertainty, channel matrix uncertainty, noise covariance uncertainty, impulse noise (outliers), and limited pilot sample size.

Proposed method

• Formulates beamforming and signal estimation as a distributionally robust optimization problem over an ambiguity set of joint signal and noise distributions.
• Uses an F-norm-based uncertainty set to quantify distributional perturbations in the signal and noise covariance, enabling robustness against model uncertainty.
• Derives optimal linear estimators via a regularized least-squares formulation, proving that ridge regression is distributionally robust under diagonal perturbations in feature covariance.
• Generalizes to nonlinear estimation by restricting estimators to reproducing kernel Hilbert spaces (RKHS), leading to kernelized diagonal loading and kernel ridge regression as special cases.
• Proposes a distributionally robust Wiener beamformer (Wnr-DR) using nontrivial F-norm uncertainty sets, outperforming standard diagonal loading in theory and practice.
• Employs cross-validation for hyperparameter tuning (e.g., ε in diagonal loading) to balance robustness and performance, avoiding over-conservatism or under-regularization.

Experimental results

Research questions

• RQ1 How can we unify classical linear beamformers (e.g., Wiener, Capon, ZF) and modern data-driven estimators (e.g., kernel, neural networks) under a single robust learning framework?
• RQ2 What is the theoretical connection between ridge regression and distributional robustness in the presence of signal and noise covariance uncertainty?
• RQ3 Can we extend the robustness of diagonal loading to nonlinear estimators and improve performance under limited pilot data and outliers?
• RQ4 Under what conditions is channel estimation actually necessary, and when can it be bypassed via direct distributionally robust estimation?
• RQ5 How do different uncertainty quantification strategies (e.g., trivial vs. nontrivial F-norm sets) affect the performance and computational cost of beamformers?

Key findings

• The Wiener-DR beamformer using a nontrivial F-norm uncertainty set achieves an MSE of 38.18 in Scenario 2 (pilot size = 20), outperforming the standard Wiener beamformer (MSE = 41.42) and Wiener-DL (MSE = 41.29).
• Kernel-based beamformers (MSE = 37.20) and kernel-DL (MSE = 34.87) significantly outperform linear beamformers under impulse noise and non-Gaussian signal models, demonstrating the advantage of nonlinear robustness.
• Diagonally loaded beamformers (e.g., Wnr-DL, Capon-DL, Kernel-DL) consistently outperform their non-diagonally loaded counterparts across all scenarios, confirming the robustness of diagonal loading against data scarcity and uncertainty.
• The Wiener-DR beamformer has higher computational cost (3.51 seconds) than Wiener-DL (2.60e-5 seconds), indicating a trade-off between robustness and efficiency, making Wiener-DL more practical for resource-constrained systems.
• Ridge and kernel ridge regression are proven to be distributionally robust under diagonal perturbations in feature covariance, providing theoretical grounding for their use in uncertain wireless environments.
• Neural network-based beamformers are more expressive but significantly more computationally expensive and harder to tune than the proposed 11 beamformers, limiting their scalability in large-scale antenna arrays.