Unsupervised Deep Equilibrium Model Learning for Large-Scale Channel Estimation with Performance Guarantees

TL;DR

This paper introduces an unsupervised deep equilibrium model for large-scale channel estimation that leverages GSURE for loss calculation, providing performance guarantees without needing ground-truth channels.

Contribution

It proposes a novel unsupervised learning framework using DEQ models and GSURE, with theoretical performance guarantees and improved results over baselines.

Findings

Outperforms baselines without ground-truth channels

Provides theoretical guarantees for MSE performance

Ensures compressible solutions via DEQ architecture

Abstract

Supervised deep learning methods have shown promise for large-scale channel estimation (LCE), but their reliance on ground-truth channel labels greatly limits their practicality in real-world systems. In this paper, we propose an unsupervised learning framework for LCE that does not require ground-truth channels. The proposed approach leverages Generalized Stein's Unbiased Risk Estimate (GSURE) as a principled unsupervised loss function, which provides an unbiased estimate of the projected mean-squared error (PMSE) from compressed noisy measurements. To ensure a guaranteed performance, we integrate a deep equilibrium (DEQ) model, which implicitly represents an infinite-depth network by directly learning the fixed point of a parameterized iterative process. We theoretically prove that, under mild conditions, the proposed GSURE-based unsupervised DEQ learning can achieve oracle-level supervised performance. In particular, we show that the DEQ architecture inherently enforces a compressible solution. We then demonstrate that DEQ-induced compressibility ensures that optimizing the projected error via GSURE suffices to guarantee a good MSE performance, enabling a rigorous performance guarantee. Extensive simulations validate the theoretical findings and demonstrate that the proposed framework significantly outperforms various baselines when ground-truth channel is unavailable.