AMP-Inspired Deep Networks for Sparse Linear Inverse Problems

TL;DR

This paper introduces two deep neural network architectures inspired by AMP algorithms for sparse linear inverse problems, improving robustness and interpretability, with applications in wireless communications.

Contribution

The paper proposes learned AMP and VAMP neural networks that incorporate Onsager correction, enhancing sparse signal recovery and robustness to measurement matrix deviations.

Findings

Learned VAMP matches theoretical AMP values for i.i.d. signals.

Both networks outperform previous methods like learned ISTA.

Applications demonstrate effectiveness in 5G wireless tasks.

Abstract

Deep learning has gained great popularity due to its widespread success on many inference problems. We consider the application of deep learning to the sparse linear inverse problem, where one seeks to recover a sparse signal from a few noisy linear measurements. In this paper, we propose two novel neural-network architectures that decouple prediction errors across layers in the same way that the approximate message passing (AMP) algorithms decouple them across iterations: through Onsager correction. First, we propose a "learned AMP" network that significantly improves upon Gregor and LeCun's "learned ISTA." Second, inspired by the recently proposed "vector AMP" (VAMP) algorithm, we propose a "learned VAMP" network that offers increased robustness to deviations in the measurement matrix from i.i.d. Gaussian. In both cases, we jointly learn the linear transforms and scalar nonlinearities of the network. Interestingly, with i.i.d. signals, the linear transforms and scalar nonlinearities prescribed by the VAMP algorithm coincide with the values learned through back-propagation, leading to an intuitive interpretation of learned VAMP. Finally, we apply our methods to two problems from 5G wireless communications: compressive random access and massive-MIMO channel estimation.