As a 5G service category, ultra-reliable low-latency communication (URLLC) raises the challenge of dramatically improving the reliability of short message transmission, for which sparse regression codes (SRCs) and their variations have emerged as promising solutions. In this paper, we propose a generalization of single-section SRCs (SRCl) by designing a sparse vector set that satisfies a certain minimum Euclidean distance constraint. The design problem is first transformed into a constant weight code (CWC) design problem. By extending the binary alphabet to an$M$-ary-phase alphabet, we generalize the CWC to$M$-ary CWC ($M$-CWC) to further increase the achievable minimum Euclidean distance. The increment is theoretically analyzed, and the anticipated performance gain of the resultant$M$-CWC-SRCI over SRCI is verified by simulation. Additionally, our simulation results show that the proposed$M$-CWC-SRCI outperforms the state-of-the-art SRCl-based schemes by about 1 dB gain in EblNo at BLER of Le - 5.

This paper investigates a large unitarily invariant system (LUIS) involving a unitarily invariant sensing matrix, an arbitrarily fixed signal distribution, and forward error control (FEC) coding. A universal Gram-Schmidt orthogonalization is considered for constructing orthogonal approximate message passing (OAMP), enabling its applicability to a wide range of prototypes without the constraint of differentiability. We develop two single-input-single-output variational transfer functions for OAMP with Lipschitz continuous local estimators, facilitating an analysis of achievable rates. Furthermore, when the state evolution of OAMP has a unique fixed point, we reveal that OAMP can achieve the constrained capacity predicted by the replica method of LUIS based on matched FEC coding, regardless of the signal distribution. The replica method is rigorously validated for LUIS with Gaussian signaling and certain sub-classes of LUIS with arbitrary signal distributions. Several area properties are established based on the variational transfer functions of OAMP. Meanwhile, we present a replica constrained capacity-achieving coding principle for LUIS. This principle serves as the basis for optimizing irregular low-density parity-check (LDPC) codes specifically tailored for binary signaling in our simulation results. The performance of OAMP with these optimized codes exhibits a remarkable improvement over the unoptimized codes and even surpasses the well-known Turbo-LMMSE algorithm. For quadrature phase-shift keying (QPSK) modulation, we observe bit error rates (BER) performance near the replica constrained capacity across diverse channel conditions.

We study the performance of a Bayesian statistician who estimates a rank-one signal corrupted by non-symmetric rotationally invariant noise with a generic distribution of singular values. As the signal-to-noise ratio and the noise structure are unknown, a Gaussian setup is incorrectly assumed. We derive the exact analytic expression for the error of the mismatched Bayes estimator and also provide the analysis of an approximate message passing (AMP) algorithm. The first result exploits the asymptotic behavior of spherical integrals for rectangular matrices and of low-rank matrix perturbations; the second one relies on the design and analysis of an auxiliary AMP. The numerical experiments show that there is a performance gap between the AMP and Bayes estimators, which is due to the incorrect estimation of the signal norm.

Sparse regression codes (SPARCs) are a promising coding scheme that can approach the Shannon limit over Additive White Gaussian Noise (AWGN) channels. Previous works have proven the capacity-achieving property of SPARCs with Gaussian design matrices. We generalize these results to right orthogonally invariant ensembles that allow for more structured design matrices. With the Vector Approximate Message Passing (VAMP) decoder, we rigorously demonstrate the exponentially decaying error probability for design matrices that satisfy a certain criterion with the exponentially decaying power allocation. For other spectra, we design a new power allocation scheme to show that the information theoretical threshold is achievable.

This paper presents a low-complexity lossy compression scheme for Gaussian vectors, using sparse regression codes (SRC) and a novel decimated approximate message passing (AMP) encoder. The sparse regression codebook is characterized by a design matrix and each codeword is a linear combination of selected columns of the matrix. In order to enable the convergence of AMP for lossy compression, we incorporate the concept of decimation into the AMP algorithm for the first time. Further, we show that the power allocation technique is beneficial for improving the rate-distortion performance. The computational complexity of the proposed encoding is O(log n) per source sample for a length-n source vector, using a sub-Fourier design matrix. Moreover, the proposed AMP encoder inherently supports successively refinable compression. Simulation results show that the proposed decimated AMP encoder significantly outperforms the existing successive-approximation encoding [1] and approaches the rate-distortion limit in low-rate regime.

We consider the problem of reconstructing the signal and the hidden variables from observations coming from a multi-layer network with rotationally invariant weight matrices. The multi-layer structure models inference from deep generative priors, and the rotational invariance imposed on the weights generalizes the i.i.d. Gaussian assumption by allowing for a complex correlation structure, which is typical in applications. In this work, we present a new class of approximate message passing (AMP) algorithms and give a state evolution recursion which precisely characterizes their performance in the large system limit. In contrast with the existing multi-layer VAMP (ML-VAMP) approach, our proposed AMP – dubbed multilayer rotationally invariant generalized AMP (ML-RI-GAMP) – provides a natural generalization beyond Gaussian designs, in the sense that it recovers the existing Gaussian AMP as a special case. Furthermore, ML-RI-GAMP exhibits a significantly lower complexity than ML-VAMP, as the computationally intensive singular value decomposition is replaced by an estimation of the moments of the design matrices. Finally, our numerical results show that this complexity gain comes at little to no cost in the performance of the algorithm.

Approximate Message Passing (AMP) is an efficient iterative parameter-estimation technique for certain high-dimensional linear systems with non-Gaussian distributions, such as sparse systems. In AMP, a so-called Onsager term is added to keep estimation errors approximately Gaussian. Orthogonal AMP (OAMP) does not require this Onsager term, relying instead on an orthogonalization procedure to keep the current errors uncorrelated with (i.e., orthogonal to) past errors. In this paper, we show the generality and significance of the orthogonality in ensuring that errors are “asymptotically independently and identically distributed Gaussian” (AIIDG). This AIIDG property, which is essential for the attractive performance of OAMP, holds for separable functions. We present a simple and versatile procedure to establish the orthogonality through Gram-Schmidt (GS) orthogonalization, which is applicable to any prototype. We show that different AMP-type algorithms, such as expectation propagation (EP), turbo, AMP and OAMP, can be unified under the orthogonal principle. The simplicity and generality of OAMP provide efficient solutions for estimation problems beyond the classical linear models. As an example, we study the optimization of OAMP via the GS model and GS orthogonalization. More related applications will be discussed in a companion paper where new algorithms are developed for problems with multiple constraints and multiple measurement variables.

In this paper, we study a concatenate coding scheme based on sparse regression code (SPARC) and tree code for unsourced random access in massive multiple-input and multiple-output systems. Our focus is concentrated on efficient decoding for the inner SPARC with practical concerns. A two-stage method is proposed to achieve near-optimal performance while maintaining low computational complexity. Specifically, a one-step thresholding-based algorithm is first used for reducing large dimensions of the SPARC decoding, after which a relaxed maximum-likelihood estimator is employed for refinement. Adequate simulation results are provided to validate the near-optimal performance and the low computational complexity. Besides, for covariance-based sparse recovery method, theoretical analyses are given to characterize the upper bound of the number of active users supported when convex relaxation is considered, and the probability of successful dimension reduction by the one-step thresholding-based algorithm.

In this paper, we study a concatenate coding scheme based on sparse regression code (SPARC) and tree code for unsourced random access in massive multiple-input and multiple-output systems. Our focus is concentrated on efficient decoding for the inner SPARC with practical concerns. A two-stage method is proposed to achieve near-optimal performance while maintaining low computational complexity [1]. Specifically, a one-step thresholding-based algorithm is first used for reducing large dimensions of the SPARC decoding, after which a relaxed maximum-likelihood estimator is employed for refinement. Adequate simulation results are provided to validate the near-optimal performance and the low computational complexity. Besides, for covariance-based sparse recovery method, theoretical analyses are given to characterize the upper bound of the number of active users supported when convex relaxation is considered, and the probability of successful dimension reduction by the one-step thresholding-based algorithm.

This paper investigates a large unitarily invariant system (LUIS) involving a unitarily invariant sensing matrix, an arbitrary fixed signal distribution, and forward error control (FEC) coding. Several area properties are established based on the state evolution of orthogonal approximate message passing (OAMP) in an un-coded LUIS. Under the assumptions that the state evolution for joint OAMP and FEC decoding is correct and the replica method is reliable, we analyze the achievable rate of OAMP. We prove that OAMP reaches the constrained capacity predicted by the replica method of the LUIS with an arbitrary signal distribution based on matched FEC coding. Meanwhile, we elaborate a constrained capacity-achieving coding principle for LUIS, based on which irregular low-density parity-check (LDPC) codes are optimized for binary signaling in the simulation results. We show that OAMP with the optimized codes has significant performance improvement over the un-optimized ones and the well-known Turbo linear MMSE algorithm. For quadrature phase-shift keying (QPSK) modulation, constrained capacity-approaching bit error rate (BER) performances are observed under various channel conditions.

This paper considers a concatenated coding scheme of sparse regression codes (SPARC) and tree code for asynchronous uplink unsourced random access. The encoding of SPARC is limited in a single sub-carrier to counter the unknown delay due to asynchronization. A slotted structure, where both channel uses and potential users are divided into slots, is added to reduce the overall computational complexity and improve its reliability when the length of the coherence block is limited. An efficient two-stage decoder is applied for further computational complexity reduction at the cost of slightly higher per-user probability of error. Asymptotic and numerical results are provided to demonstrate the effectiveness of the proposed scheme.

This paper considers a compressed-coding scheme that combines compressed sensing with forward error control coding. Approximate message passing (AMP) is used to decode the message. Based on the state evolution analysis of AMP, we derive the performance limit of compressed-coding. We show that compressed-coding can approach Gaussian capacity at a very low compression ratio. Further, the results are extended to systems involving non-linear effects such as clipping. We show that the capacity approaching property can still be maintained when generalized AMP is used to decode the message. To approach the capacity, a low-rate underlying code should be designed according to the curve matching principle, which is complicated in practice. Instead, analog spatial-coupling is used to avoid sophisticated low-rate code design.

This paper concerns practical issues of sparse regression codes (SRCs) with approximate message passing (AMP) decoding. First, Gaussian signaling of SRC incurs a high peak-to-average-power ratio (PAPR) problem. Second, the finite length performance of SRCs is poor at low-to-medium rates and cannot be improved by spatial coupling or power allocation. We confront the two challenges by introducing clipping to SRC. For the encoder, clipping is applied to the Gaussian codeword of SRC for reducing the high PAPR. For the decoder, generalized approximate message passing (GAMP) is used to handle the nonlinear clipping distortion. Interestingly, we observe that clipping with proper thresholds can improve the performance of SRC and the performance gain is large at low rates. Based on the state evolution analysis of GAMP decoding, we provide an explanation for such observation from the curve-matching perspective. In the end, some guidelines are provided for choosing proper clipping thresholds empirically.

This paper considers a compressed-coding scheme that combines compressed sensing with forward error control coding. Approximate message passing (AMP) is used to decode the message. Based on the state evolution analysis of AMP, we derive the performance limit of compressed-coding. We show that compressed-coding can approach Gaussian capacity at a very low compression ratio. Further, the results are extended to systems involving non-linear effects such as clipping. We show that the capacity approaching property can still be maintained when generalized AMP is used to decode the message. To approach the capacity, a low-rate underlying code should be designed according to the curve matching principle, which is complicated in practice. Instead, analog spatial-coupling is used to avoid sophisticated low-rate code design. In the end, we study the coupled scheme in a multiuser environment, where analog spatial-coupling can be realized in a distributive way. The overall block length can be shared by many users, which reduces block length per-user.

In massive multiple-input multiple-output (MIMO) systems with hybrid analog/digital architectures, large training overhead is required for conventional pilot-only methods to estimate channel accurately before detecting data. To reduce the training overhead, a semi-blind detection method is proposed for data detection without knowing channel in an uplink multi-user system. The main idea is to exploit the received signal corresponding to both the pilot and data payload for channel estimation or data detection via a low-rank matrix completion formulation. The leveraged low-rank property stems from the fact that the number of active users K is typically much smaller than the number of antennas Naat a base station and the number of time slots Tcin a coherence interval. Compared with the pilot-only method, the number of pilots required is reduced from an order of Nato K. Two iterative algorithms are introduced to solve the low-rank matrix completion problem: regularized alternating least squares and bilinear generalized approximate message passing. We further extend the semi-blind detection method to systems with low-resolution analog-to-digital converters. Simulation results show that the proposed methods achieve significant performance gain over the pilot-only method with reduced training overhead for hybrid massive MIMO systems in various settings.