Vector Approximate Message Passing Research Articles

Time domain turbo equalization based on vector approximate message passing for multiple-input multiple-output underwater acoustic communications.

This paper proposes a high-performance receiver for underwater acoustic communications based on time reversal processing for multiple-input multiple-output (MIMO) systems. The receiver employs the vector approximate message passing (VAMP) algorithm as a soft equalizer in turbo equalization. By performing self-iteration between the inner soft slicer and the inner soft equalizer, the VAMP algorithm achieves near-optimal performance. Furthermore, an iterative channel-estimation-based soft successive interference cancellation method is incorporated to suppress co-channel interference in the MIMO system. Additionally, the introduction of passive time reversal technology can combine multiple channels into a single channel, which greatly reduces the computational complexity of the MIMO system, especially for large MIMO systems. The effectiveness of the proposed receiver is verified using experimental data collected in Songhua Lake, China in 2019. The results demonstrate that the proposed receiver significantly reduces the complexity of the traditional parallel-VAMP receiver without sacrificing performance and outperforms other receivers of the same type. Moreover, our experimental results also verify that the VAMP-turbo outperforms the generalized approximate message passing (GAMP)-turbo in terms of bit error rate and convergence performance.

Vector Approximate Message Passing Based OFDM-IM Detection for Underwater Acoustic Communications.

Orthogonal frequency division multiplexing with index modulation (OFDM-IM) has great potential for the implementation of high spectral-efficiency underwater acoustic (UWA) communications. However, general receivers consisting of the optimal maximum likelihood detection suffer from high computational load, which prohibits real-time data transmissions in underwater scenarios. In this paper, we propose a detection based on a vector approximate message passing (VAMP) algorithm for UWA OFDM-IM communications. Firstly, a VAMP framework with a non-loopy factor graph for index detection is formulated. Secondly, by utilizing the sparsity inherently existing in OFDM-IM symbols, a novel shrinkage function is derived based on the minimum mean square error criterion, which guarantees better posterior estimation. To reduce the errors from estimated non-existing indices, one trick is utilized to search the elements from the look-up table with the minimal Euclidean distance for the replacement of erroneously estimated indices. Experiments verify the advantages of the proposed detector in terms of low complexity, robustness and effectiveness compared with the state-of-art benchmarks.

Expectation-maximization vector approximate message passing-based frequency-domain turbo equalization for underwater acoustic communications.

Channel equalization plays a crucial role in single-carrier underwater acoustic (UWA) communications. Recently, a frequency-domain turbo equalization (FDTE) scheme enabled by the vector approximate message passing (VAMP) algorithm, was proposed, and it outperformed classic linear minimum mean square error FDTE at acceptable complexity cost. The operation of the VAMP-FDTE requires knowledge of noise power, which is predetermined before the equalization starts. In practice, however, it is difficult to obtain prior knowledge of noise power due to factors of unknown channel estimation errors and dynamic underwater environments. Motivated by this fact, we propose an enhanced VAMP-FDTE scheme, which learns the noise power knowledge during the equalization process via the expectation-maximization (EM) algorithm. The EM-based noise power estimation makes use of intermediate results of the VAMP-FDTE and, thus, only incurs a small extra computational overhead. The improved VAMP-FDTE, named EM-VAMP-FDTE, was tested by experimental data collected in shallow-sea horizontal UWA communication trials with MIMO configuration. It showed better performance than the existing VAMP-FDTE scheme, attributed to the online noise power learning.

Probabilistic constellation shaping-aided underwater acoustic communication with vector approximate message passing iterative equalization.

A probabilistic constellation shaping (PCS)-aided single-carrier transceiver is proposed to improve spectral efficiency for underwater acoustic (UWA) communications. At the transmitter, the information bits are input into a distribution matcher followed by a systematic binary encoder, which yields a sequence of sign bits with a uniform distribution and a sequence of amplitude bits with a non-uniform distribution. Based on these two sequences, the PCS is then realized by mapping coded bits onto a quadrature amplitude modulation constellation. At the receiver, an improved frequency-domain turbo equalizer based on the vector approximate message passing (VAMP) is proposed for the PCS-UWA communication system to eliminate the multipath interference. It exploits the a priori symbol probability information benefiting from the PCS at the beginning of the turbo iteration and over the self-iteration of the VAMP soft equalizer, improving the symbol detection performance. Finally, the first experimental demonstration of a deep-sea PCS-UWA communication system and numerical simulation of shallower water are presented. The experimental results reveal the PCS-UWA communication system significantly outperforms traditional systems with no PCS. Also, the proposed receiver is superior to the classical adaptive turbo equalizer based on the improved proportionate normalized least mean square algorithm even with data reuse.

Vector approximate message passing with sparse Bayesian learning for Gaussian mixture prior

Compressed sensing (CS) aims for seeking appropriate algorithms to recover a sparse vector from noisy linear observations. Currently, various Bayesian-based algorithms such as sparse Bayesian learning (SBL) and approximate message passing (AMP) based algorithms have been proposed. For SBL, it has accurate performance with robustness while its computational complexity is high due to matrix inversion. For AMP, its performance is guaranteed by the severe restriction of the measurement matrix, which limits its application in solving CS problem. To overcome the drawbacks of the above algorithms, in this paper, we present a low complexity algorithm for the single linear model that incorporates the vector AMP (VAMP) into the SBL structure with expectation maximization (EM). Specifically, we apply the variance auto-tuning into the VAMP to implement the E step in SBL, which decrease the iterations that require to converge compared with VAMP-EM algorithm when using a Gaussian mixture (GM) prior. Simulation results show that the proposed algorithm has better performance with high robustness under various cases of difficult measurement matrices.

VAMP-Based Iterative Equalization for Index-Modulated Multicarrier FTN Signaling

The spectrally efficient multicarrier faster-than-Nyquist (MFTN) signaling provides an efficient and robust solution for enhancing transmission rate and resisting channel impairments. In this paper, an evolutionary non-orthogonal physical waveform is proposed for simultaneously achieving high spectral and energy efficiency via combining MFTN signaling with index modulation (IM). Exploiting the implicit transmission of IM, the inactivated subcarriers alleviate the inherent two-dimensional interferences imposed by time-frequency packing in MFTN. Then, we develop a pair of iterative equalization algorithms based on vector approximate message passing (VAMP). For the first time-domain equalization (TDE), an extended constellation set is constructed for uniformly characterizing the activated and inactivated subcarriers. To further reducing the computational complexity, we establish a subcarrier-based segment-wise frequency-domain received signal model and accordingly develop low-complexity VAMP-based frequency-domain equalization (FDE) combined with interference elimination. Corroborated by simulations, the novel MFTN-IM waveform achieves superior bit error rate (BER) performance over benchmark waveforms in the high signal noise ratio (SNR) regions. Interestingly, while the proposed VAMP-TDE outperforms the proposed VAMP-FDE in BER performance, the latter is more competitive in balancing demodulation performance and computational complexity.

UAMPnet: Unrolled approximate message passing network for nonconvex regularization

Deep neural networks and model-based methods are both popular for their wide and great success in many inference problems. In this paper, resorting to deep learning, we study the efficient algorithms for two popular nonconvex regularization methods, smoothly clipped absolute deviation (SCAD) and minimax concave penalty (MCP). Approximate message passing (AMP) algorithm can be effective to optimize nonconvex regularization models. First, we unroll the AMP-based algorithm as the feed-forward neural network through leveraging the novel activation functions of neurons, dubbed as Unrolled-AMP. And then, for the case where the measurement matrix deviates from the i.i.d. Gaussian distribution, we propose two improved iterative algorithms based on the “Vector AMP (VAMP)” algorithm to solve the nonconvex regularization methods. Further, we unroll them as the feed-forward neural network, dubbed as Unrolled-VAMP. These two novel neural network architectures use a back-propagation algorithm to learn all their parameters from the training data. Finally, the convergence of Unrolled-AMP algorithm is analyzed, and the efficiency of the proposed networks is demonstrated through experiments on sparse signal reconstruction and 5G wireless communication.

Macroscopic Analysis of Vector Approximate Message Passing in a Model-Mismatched Setting

In this study, macroscopic properties of the vector approximate message passing (VAMP) algorithm for inference of generalized linear models are investigated using a non-rigorous heuristic method of statistical mechanics when the true posterior cannot be used and the measurement matrix is a sample from rotation-invariant random matrix ensembles. The focus is on the correspondence between the non-rigorous replica analysis of statistical mechanics and the performance assessment of VAMP in the model-mismatched setting. The correspondence of this kind is well-known when the measurement matrix has independent and identically distributed entries. However, when the measurement matrix follows a general rotation-invariant matrix ensemble, the correspondence has been validated only under limited cases, such as the Bayes optimal inference or the convex empirical risk minimization. The result presented in this paper is to extend the scope of such correspondence. Herein, we heuristically derive the explicit formula of state-evolution equations, which macroscopically describe VAMP dynamics for the current model-mismatched case, and show that their fixed point is generally consistent with the replica symmetric solution obtained by the replica method of statistical mechanics. We also show that the fixed point of VAMP can exhibit a microscopic instability, which indicates that message variables continue to move by VAMP while their macroscopically summarized quantities converge to fixed values. The critical condition the for microscopic instability agrees with that for breaking the replica symmetry that is derived within the non-rigorous replica analysis. The results of the numerical experiments cross-check our findings.

Approximate Message Passing for Channel Estimation in Reconfigurable Intelligent Surface Aided MIMO Multiuser Systems

Channel estimation is one of the main challenges for implementing reconfigurable intelligent surface (RIS)-aided communication system with a large number of antennas at the base station (BS) because RIS is equipped with many passive reflective elements without active transmitting/receiving and signal processing abilities. In this paper, we focus on the channel estimation in a general RIS-aided multi-user mmWave communication system. Specifically, a novel two-phase channel estimation scheme consisting of on-line and off-line phases is proposed to estimate the BS-RIS channel, RIS-user channel, and BS-user channel, respectively. Inspired by the characteristics of few scatterers in mmWave communication systems, the antenna domain channels are converted into the virtual angular domain channels by using the discrete Fourier transform (DFT) matrix. Thus, the estimation problem can be formulated as a compressed sensing (CS) problem, and solved by using efficient vector approximate message passing (VAMP) algorithm with expectation-maximization (EM) to learn unknown parameters and obtain the estimates simultaneously. Simulation results show that, the above algorithm with two choices of prior distributions in the proposed mmWave RIS-aided multi-user system has fast convergence speed and desirable estimation performance.

Compressed Sensing With Upscaled Vector Approximate Message Passing

The Recently proposed Vector Approximate Message Passing (VAMP) algorithm demonstrates a great reconstruction potential at solving compressed sensing related linear inverse problems. VAMP provides high per-iteration improvement, can utilize powerful denoisers like BM3D, has rigorously defined dynamics and is able to recover signals measured by highly undersampled and ill-conditioned linear operators. Yet, its applicability is limited to relatively small problem sizes due to the necessity to compute the expensive LMMSE estimator at each iteration. In this work we consider the problem of upscaling VAMP by utilizing Conjugate Gradient (CG) to approximate the intractable LMMSE estimator. We propose a rigorous method for correcting and tuning CG withing CG-VAMP to achieve a stable and efficient reconstruction. To further improve the performance of CG-VAMP, we design a warm-starting scheme for CG and develop theoretical models for the Onsager correction and the State Evolution of Warm-Started CG-VAMP (WS-CG-VAMP). Additionally, we develop robust and accurate methods for implementing the WS-CG-VAMP algorithm. The numerical experiments on large-scale image reconstruction problems demonstrate that WS-CG-VAMP requires much fewer CG iterations compared to CG-VAMP to achieve the same or superior level of reconstruction.

A Low-Complexity Joint Estimation and Detection Scheme of OFDM System With Combined BP-MF and VAMP

Considering orthogonal frequency division multiplexing (OFDM) system, we propose a joint channel estimation, noise precision estimation, symbol detection and bit decoding algorithm. In this algorithm, we incorporate the recently proposed vector approximate message passing (VAMP) with the belief propagation and mean field (BP-MF) framework, where BP is applied within subgraph of demodulation and decoding, and MF is used to handle nonlinear observations for estimating unknown noise precision, and VAMP is employed to estimate channels in densely connected factor graph. Furthermore, we use expectation propagation (EP) to demonstrate that the proposed approach converges after only one iteration. Compared to the state-of-the-art algorithm which exploits generalized approximate message passing (GAMP), the proposed scheme achieves the same bit error ratio (BER) performance, but lower per-iteration complexity and faster convergence speed.

Joint estimation and detection method based on turbo equalization framework and VAMP

In this letter, we consider the single-carrier frequency domain equalization (SC-FDE) system, and propose a low-complexity joint symbol detection and channel estimation algorithm based on the recently proposed vector approximate message passing (VAMP). Specifically, we leverage VAMP twice to estimate symbols and channels, respectively, in a turbo-like way. Moreover, this algorithm organically combines the gaussian mixture model (GMM), which can accurately simulate the sparse aggregation characteristics of the channel and effectively suppress inter symbol interference (ISI). The simulation results show that compared with the traditional linear minimum mean square error (LMMSE) estimation receiving algorithm and the existing generalized approximate message passing algorithm (GAMP), the designed receiving algorithm has significant advantages in channel estimation normalized mean square error (NMSE) and bit error ratio (BER) performance, where sharing the same order of complexity.

Joint Active and Passive Beamforming Design for IRS-Assisted Multi-User MIMO Systems: A VAMP-Based Approach

This paper tackles the problem of joint active and passive beamforming optimization for an intelligent reflective surface (IRS)-assisted multi-user downlink multiple-input multiple-output (MIMO) communication system under both ideal and practical IRS phase shifts. We aim to maximize the spectral efficiency of the users by minimizing the sum mean square error (MSE) of the users’ received symbols. For this, a joint non-convex optimization problem is formulated under the sum minimum mean square error (MMSE) criterion. Alternating minimization is used to break the original joint optimization problem into the separate optimization of the active precoding matrix for the base station (BS) and the matrix of phase shifts for the IRS. While the MMSE active precoder is obtained in closed-form, the IRS phase shifts are optimized iteratively using a modified version (developed in this paper) of the vector approximate message passing (VAMP) algorithm. Moreover, the underlying joint optimization problem is solved under two different models for the IRS phase shifts, namely by assuming <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> ) a unimodular (i.e., ideal) constraint on the reflection coefficients and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ii$ </tex-math></inline-formula> ) a more practical reflection elements termination by a variable reactive load (which inherently introduces the phase-dependent amplitude attenuation in the IRS phase shifts). Simulation results are presented to illustrate the performance of the proposed method under both perfect and imperfect channel state information (CSI) and to show the effect of the practical constraint on the system throughput. The results validate the superiority of the proposed method over the state-of-the-art techniques both in terms of throughput and computational complexity.

An enhanced iterative receiver based on vector approximate message passing for deep-sea vertical underwater acoustic communications.

Vertical underwater acoustic (UWA) communications play a crucial role in deep-sea applications. A vertical UWA channel generally features a moderate multipath but with time-varying Doppler shifts as well as loud impulsive noise. To achieve a robust vertical single-carrier UWA communication, this paper proposes an enhanced iterative receiver. First, a spline interpolation-based timing estimation approach is proposed to compensate for the time-varying Doppler effects efficiently. Then, the residual timing errors and the multipath interference are tackled by a fractionally spaced self-iterative soft equalizer (SISE) based on the vector approximate message passing (VAMP) algorithm. The VAMP-SISE consists of four parts: an inner soft slicer and an inner soft equalizer for symbol detection as well as a denoiser and a minimum mean-squared error estimator for impulsive noise suppression. Different parts iteratively exchange extrinsic information to improve the equalization performance. Last, a channel-fitting irregular convolutional code and a unity-rate code are employed at the transmitter to lower the signal-to-noise ratio threshold for reliable communications. Deep-sea experiments verify the performance superiority of the proposed receiver over existing schemes.

Advanced NOMA Receivers From a Unified Variational Inference Perspective

Non-orthogonal multiple access (NOMA) on shared resources has been identified as a promising technology in 5G to improve resource efficiency and support massive access in all kinds of transmission modes. Power domain and code domain NOMA have been extensively studied and evaluated in both literatures and 3GPP standardization, especially for the uplink where large number of users would like to send their messages to the base station. Though different in the transmitter side design, power domain NOMA and code domain NOMA share the same need of the advanced multi-user detection (MUD) design at the receiver side. Various multi-user detection algorithms have been proposed, balancing performance and complexity in different ways, which is important for the implementation of NOMA in practical networks. In this paper, we introduce a unified variational inference (VI) perspective on various universal NOMA MUD algorithms such as belief propagation (BP), expectation propagation (EP), vector EP (VEP), approximate message passing (AMP) and vector AMP (VAMP), demonstrating how they could be derived from and adapted to each other within the VI framework. Moreover, we unveil and prove that conventional elementary signal estimator (ESE) and linear minimum mean square error (LMMSE) receivers are special cases of EP and VEP, respectively, thus bridging the gap between classic linear receivers and message passing based nonlinear receivers. Such a unified perspective would not only help the design and adaptation of NOMA receivers, but also open a door for the systematic design of joint active user detection and multi-user decoding for sporadic grant-free transmission.

Near-Optimal Self-Iterative VAMP Equalization Enabled by Hadamard-Haar Random Precoding

A self-iterative vector approximate message passing (VAMP) soft frequency-domain equalizer (SFDE) is a preferred choice for turbo equalization due to its excellent complexity-performance tradeoff. To maximize its performance, the equivalent channel matrix is required to be right-rotationally invariant (RRI) which generally does not hold though. As a result, existing VAMP-SFDEs cannot reach the performance bound. To solve such a dilemma, we propose in this letter to introduce a precoding operation on the transmitter side such that a RRI equivalent channel matrix can be obtained. The designed precoding scheme is named the Hadamard-Haar random precoding (HHRP), which is the concatenation of an inverse fast Fourier transform matrix, a random interleaving matrix, and a Hadamard-Haar transform (HHT) matrix. Attributed to the HHRP, the VAMP-SFDE applied in turbo equalization achieves extra performance gain and enjoys faster convergence over existing VAMP-SFDEs.

A Generalized VAMP-Based Channel Estimator for Uplink Quantized Massive MIMO Systems

In this study, we propose an improved channel estimator based on vector approximate message passing (VAMP) for quantized millimeter wave (mmWave) massive multiple-input and multiple-output (MIMO) systems. Using channel sparsity and the orthogonality of steering matrix, the proposed VAMP-on method designed for the on-grid scenario not only improves estimation performance but also has better convergence and inverse-free implementation. Furthermore, to achieve better estimation performance under the off-grid scenario, we provide a VAMP-off algorithm to update steering matrix and channel coefficients in an alternating fashion. Finally, simulation results are provided to demonstrate the advantages of the proposed approaches.

Inference With Deep Generative Priors in High Dimensions

Deep generative priors offer powerful models for complex-structured data, such as images, audio, and text. Using these priors in inverse problems typically requires estimating the input and/or hidden signals in a multi-layer deep neural network from observation of its output. While these approaches have been successful in practice, rigorous performance analysis is complicated by the non-convex nature of the underlying optimization problems. This paper presents a novel algorithm, Multi-Layer Vector Approximate Message Passing (ML-VAMP), for inference in multi-layer stochastic neural networks. ML-VAMP can be configured to compute maximum a priori (MAP) or approximate minimum mean-squared error (MMSE) estimates for these networks. We show that the performance of ML-VAMP can be exactly predicted in a certain high-dimensional random limit. Furthermore, under certain conditions, ML-VAMP yields estimates that achieve the minimum (i.e., Bayes-optimal) MSE as predicted by the replica method. In this way, ML-VAMP provides a computationally efficient method for multi-layer inference with an exact performance characterization and testable conditions for optimality in the large-system limit.

Plug in estimation in high dimensional linear inverse problems a rigorous analysis**This article is an updated version of: Fletcher A K, Pandit P, Rangan S, Sarkar S and Schniter P 2018 Plug-in estimation in high-dimensional linear inverse problems: a rigorous analysis Advances in Neural Information Processing Systems 31 ed S Bengio, H Wallach, H Larochelle, K Grauman, N Cesa-Bianchi and

Estimating a vector from noisy linear measurements often requires use of prior knowledge or structural constraints on for accurate reconstruction. Several recent works have considered combining linear least-squares estimation with a generic or ‘plug-in’ denoiser function that can be designed in a modular manner based on the prior knowledge about . While these methods have shown excellent performance, it has been difficult to obtain rigorous performance guarantees. This work considers plug-in denoising combined with the recently-developed vector approximate message passing (VAMP) algorithm, which is itself derived via expectation propagation techniques. It shown that the mean squared error of this ‘plug-and-play’ VAMP can be exactly predicted for high-dimensional right-rotationally invariant random and Lipschitz denoisers. The method is demonstrated on applications in image recovery and parametric bilinear estimation.

Downlink Channel Estimation for Massive MIMO Systems Relying on Vector Approximate Message Passing

To reduce the pilot overhead of downlink channel estimation in massive multiple-input-multiple-output (MIMO) systems, a sparse recovery algorithm relying on the vector approximate message passing (VAMP) technique is proposed. More specifically, an a-priori channel model characterized by a multivariate Bernoulli-Gaussian distribution is invoked for exploiting the common sparsity of massive MIMO channels, and the VAMP technique is used for jointly estimating the spatially correlated channels. Moreover, the hyperparameters of the a-priori model are learned by invoking the expectation maximization algorithm. Our numerical results demonstrate that the proposed algorithm is capable of reducing the pilot overhead by 50% in massive MIMO systems.