Sparse Regression Codes Research Articles

An Efficient Two-Stage SPARC Decoder for Massive MIMO Unsourced Random Access

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.

CRC-Aided Sparse Regression Codes for Unsourced Random Access

This letter considers a coding scheme for unsourced random access (URA) based on sparse regression codes (SPARCs). In particular, an efficient concatenated coding scheme is proposed, which concatenates SPARCs and cyclic redundancy check-based block Markov superposition transmission (CRC-BMST) codes. A hybrid decoder consisting of a successive cancellation algorithm and a simplified approximated message passing (AMP) algorithm is presented for inner SPARCs, and an improved tree decoder is proposed for outer CRC-BMST codes by introducing a pruning technique. Simulation results show the proposed coding scheme outperforms the coded compressed sensing (CCS) scheme with lower computational complexity.

A Fully Bayesian Approach for Massive MIMO Unsourced Random Access

In this paper, we propose a novel fully Bayesian approach for the massive multiple-input multiple-output (MIMO) massive unsourced random access (URA). The payload of each user device is coded by the sparse regression codes (SPARCs) without redundant parity bits. A Bayesian model is established to capture the probabilistic characteristics of the overall system. Particularly, we adopt the core idea of the model-based learning approach to establish a flexible Bayesian channel model to adapt the complex environments. Different from the traditional divide-and-conquer or pilot-based massive MIMO URA strategies, we propose a three-layer message passing (TLMP) algorithm to jointly decode all the information blocks, as well as acquire the massive MIMO channel, which adopts the core idea of the variational message passing and approximate message passing. We verify that our proposed TLMP significantly enhances the spectral efficiency compared with the state-of-the-arts baselines, and is more robust to the possible codeword collisions.

Fundamental Limits of Many-User MAC With Finite Payloads and Fading

Consider a (multiple-access) wireless communication system where users are connected to a unique base station over a shared-spectrum radio links. Each user has a fixed number k of bits to send to the base station, and his signal gets attenuated by a random channel gain (quasi-static fading). In this paper we consider the many-user asymptotics of Chen-Chen-Guo'2017, where the number of users grows linearly with the blocklength. Differently, though, we adopt a per-user probability of error (PUPE) criterion (as opposed to classical joint-error probability criterion). Under PUPE the finite energy-per-bit communication is possible, and we are able to derive bounds on the tradeoff between energy and spectral efficiencies. We reconfirm the curious behaviour (previously observed for non-fading MAC) of the possibility of almost perfect multi-user interference (MUI) cancellation for user densities below a critical threshold. Further, we demonstrate the suboptimality of standard solutions such as orthogonalization (i.e. TDMA/FDMA) and treating interference as noise (i.e. pseudo-random CDMA without multi-user detection). Notably, the problem treated here can be seen as a variant of support recovery in compressed sensing for the unusual definition of sparsity with one non-zero entry per each contiguous section of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> coordinates. This identifies our problem with that of the sparse regression codes (SPARCs) and hence our results can be equivalently understood in the context of SPARCs with sections of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">100</sup> . Finally, we discuss the relation of the almost perfect MUI cancellation property and the replica-method predictions.

Capacity-Achieving Spatially Coupled Sparse Superposition Codes With AMP Decoding

Sparse superposition codes, also called sparse regression codes (SPARCs), are a class of codes for efficient communication over the AWGN channel at rates approaching the channel capacity. In a standard SPARC, codewords are sparse linear combinations of columns of an i.i.d. Gaussian design matrix, while in a spatially coupled SPARC the design matrix has a block-wise structure, where the variance of the Gaussian entries can be varied across blocks. A well-designed spatial coupling structure can significantly enhance the error performance of iterative decoding algorithms such as Approximate Message Passing (AMP). In this paper, we obtain a non-asymptotic bound on the probability of error of spatially coupled SPARCs with AMP decoding. Applying this bound to a simple band-diagonal design matrix, we prove that spatially coupled SPARCs with AMP decoding achieve the capacity of the AWGN channel. The bound also highlights how the decay of error probability depends on each design parameter of the spatially coupled SPARC. An attractive feature of AMP decoding is that its asymptotic mean squared error (MSE) can be predicted via a deterministic recursion called state evolution. Our result provides the first proof that the MSE concentrates on the state evolution prediction for spatially coupled designs. Combined with the state evolution prediction, this result implies that spatially coupled SPARCs with the proposed band-diagonal design are capacity-achieving. Using the proof technique used to establish the main result, we also obtain a concentration inequality for the MSE of AMP applied to compressed sensing with spatially coupled design matrices. Finally we provide numerical simulation results that demonstrate the finite length error performance of spatially coupled SPARCs. The performance is compared with coded modulation schemes that use LDPC codes from the DVB-S2 standard.

SPARCs for Unsourced Random Access

Unsourced random-access (U-RA) is a type of grant-free random access with a virtually unlimited number of users, of which only a certain number $K_a$ are active on the same time slot. Users employ exactly the same codebook, and the task of the receiver is to decode the list of transmitted messages. We present a concatenated coding construction for U-RA on the AWGN channel, in which a sparse regression code (SPARC) is used as an inner code to create an effective outer OR-channel. Then an outer code is used to resolve the multiple-access interference in the OR-MAC. We propose a modified version of the approximate message passing (AMP) algorithm as an inner decoder and give a precise asymptotic analysis of the error probabilities of the AMP decoder and of a hypothetical optimal inner MAP decoder. This analysis shows that the concatenated construction can achieve a vanishing per-user error probability in the limit of large blocklength and a large number of active users at sum-rates up to the symmetric Shannon capacity, i.e. as long as $K_aR < 0.5\log_2(1+K_a\SNR)$. This extends previous point-to-point optimality results about SPARCs to the unsourced multiuser scenario. Furthermore, we give an optimization algorithm to find the power allocation for the inner SPARC code that minimizes the required $\SNR$.

Using List Decoding to Improve the Finite-Length Performance of Sparse Regression Codes

We consider sparse regression codes (SPARCs) over complex AWGN channels. Such codes can be efficiently decoded by an approximate message passing (AMP) decoder, whose performance can be predicted via so-called state evolution in the large-system limit. In this paper, we mainly focus on how to use concatenation of SPARCs and cyclic redundancy check (CRC) codes on the encoding side and use list decoding on the decoding side to improve the finite-length performance of the AMP decoder for SPARCs over complex AWGN channels. Simulation results show that such a concatenated coding scheme works much better than SPARCs with the original AMP decoder and results in a steep waterfall-like behavior in the bit-error rate performance curves. Furthermore, we apply our proposed concatenated coding scheme to spatially coupled SPARCs. Besides that, we also introduce a novel class of design matrices, i.e., matrices that describe the encoding process, based on circulant matrices derived from Frank or from Milewski sequences. This class of design matrices has comparable encoding and decoding computational complexity as well as very close performance with the commonly-used class of design matrices based on discrete Fourier transform (DFT) matrices, but gives us more degrees of freedom when designing SPARCs for various applications.

An improved analysis of least squares superposition codes with bernoulli dictionary

For the additive white Gaussian noise channel with average power constraint, sparse superposition codes (or sparse regression codes), proposed by Barron and Joseph in 2010, achieve the capacity. While the codewords of the original sparse superposition codes are made with a dictionary matrix drawn from a Gaussian distribution, we consider the case that it is drawn from a Bernoulli distribution. We show an improved upper bound on its block error probability with least squares decoding, which is fairly simplified and tighter bound than our previous result in 2014.

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Sparse superposition codes, or sparse regression codes, constitute a new class of codes, which was first introduced for communication over the additive white Gaussian noise (AWGN) channel. It has been shown that such codes are capacity-achieving over the AWGN channel under optimal maximum-likelihood decoding as well as under various efficient iterative decoding schemes equipped with power allocation or spatially coupled constructions. Here, we generalize the analysis of these codes to a much broader setting that includes all memoryless channels. We show, for a large class of memoryless channels, that spatial coupling allows an efficient decoder, based on the generalized approximate message-passing (GAMP) algorithm, to reach the potential (or Bayes optimal) threshold of the underlying (or uncoupled) code ensemble. Moreover, we argue that spatially coupled sparse superposition codes universally achieve capacity under GAMP decoding by showing, through analytical computations, that the error floor vanishes and the potential threshold tends to capacity, as one of the code parameters goes to infinity. Furthermore, we provide a closed-form formula for the algorithmic threshold of the underlying code ensemble in terms of Fisher information. Relating an algorithmic threshold to a Fisher information has theoretical as well as practical importance. Our proof relies on the state evolution analysis and uses the potential method developed in the theory of low-density parity-check (LDPC) codes and compressed sensing.

The Error Probability of Sparse Superposition Codes With Approximate Message Passing Decoding

Sparse superposition codes, or sparse regression codes (SPARCs), are a recent class of codes for reliable communication over the AWGN channel at rates approaching the channel capacity. Approximate message passing (AMP) decoding, a computationally efficient technique for decoding SPARCs, has been proven to be asymptotically capacity-achieving for the AWGN channel. In this paper, we refine the asymptotic result by deriving a large deviations bound on the probability of AMP decoding error. This bound gives insight into the error performance of the AMP decoder for large but finite problem sizes, giving an error exponent as well as guidance on how the code parameters should be chosen at finite block lengths. For an appropriate choice of code parameters, we show that for any fixed rate less than the channel capacity, the decoding error probability decays exponentially in $n/(\log n)^{2T}$, where $T$, the number of AMP iterations required for successful decoding, is bounded in terms of the gap from capacity.

The Rate-Distortion Function and Excess-Distortion Exponent of Sparse Regression Codes With Optimal Encoding

This paper studies the performance of sparse regression codes for lossy compression with the squared-error distortion criterion. In a sparse regression code, codewords are linear combinations of subsets of columns of a design matrix. It is shown that with minimum-distance encoding, sparse regression codes achieve the Shannon rate-distortion function for i.i.d. Gaussian sources $R^*(D)$ as well as the optimal excess-distortion exponent. This completes a previous result which showed that $R^*(D)$ and the optimal exponent were achievable for distortions below a certain threshold. The proof of the rate-distortion result is based on the second moment method, a popular technique to show that a non-negative random variable $X$ is strictly positive with high probability. In our context, $X$ is the number of codewords within target distortion $D$ of the source sequence. We first identify the reason behind the failure of the standard second moment method for certain distortions, and illustrate the different failure modes via a stylized example. We then use a refinement of the second moment method to show that $R^*(D)$ is achievable for all distortion values. Finally, the refinement technique is applied to Suen's correlation inequality to prove the achievability of the optimal Gaussian excess-distortion exponent.

Rateless Lossy Compression via the Extremes.

We begin by presenting a simple lossy compressor operating at near-zero rate: The encoder merely describes the indices of the few maximal source components, while the decoder's reconstruction is a natural estimate of the source components based on this information. This scheme turns out to be near optimal for the memoryless Gaussian source in the sense of achieving the zero-rate slope of its distortion-rate function. Motivated by this finding, we then propose a scheme comprised of iterating the above lossy compressor on an appropriately transformed version of the difference between the source and its reconstruction from the previous iteration. The proposed scheme achieves the rate distortion function of the Gaussian memoryless source (under squared error distortion) when employed on any finite-variance ergodic source. It further possesses desirable properties, and we, respectively, refer to as infinitesimal successive refinability, ratelessness, and complete separability. Its storage and computation requirements are of order no more than (n2)/(log β n) per source symbol for β > 0 at both the encoder and the decoder. Though the details of its derivation, construction, and analysis differ considerably, we discuss similarities between the proposed scheme and the recently introduced Sparse Regression Codes of Venkataramanan et al.

Shaped Gaussian Dictionaries for Quantized Networked Control Systems With Correlated Dropouts

This paper studies fixed rate vector quantisation for noisy networked control systems (NCSs) with correlated packet dropouts. In particular, a discrete-time linear time invariant system is to be controlled over an error-prone digital channel. The controller uses (quantized) packetized predictive control to reduce the impact of packet losses. The proposed vector quantizer is based on sparse regression codes (SPARC), which have recently been shown to be efficient in open-loop systems when coding white Gaussian sources. The dictionaries in existing design of SPARCs consist of independent and identically distributed (i.i.d.) Gaussian entries. However, we show that a significant gain can be achieved by using Gaussian dictionaries that are shaped according to the second-order statistics of the NCS in question. Furthermore, to avoid training of the dictionaries, we provide closed-form expressions for the required second-order statistics in the absence of quantization.

Lossy Compression via Sparse Linear Regression: Performance Under Minimum-Distance Encoding

We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. Called a sparse superposition or sparse regression codebook, this structure is motivated by an analogous construction proposed recently by Barron and Joseph for communication over an Additive White Gaussian Noise channel. For independent identically distributed (i.i.d) Gaussian sources and minimum-distance encoding, we show that such a code can attain the Shannon rate-distortion function with the optimal error exponent, for all distortions below a specified value. It is also shown that sparse regression codes are robust in the following sense: a codebook designed to compress an i.i.d Gaussian source of variance σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> with (squared-error) distortion D can compress any ergodic source of variance less than σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> to within distortion D. Thus, the sparse regression ensemble retains many of the good covering properties of the i.i.d random Gaussian ensemble, while having a compact representation in terms of a matrix whose size is a low-order polynomial in the block-length.

Lossy Compression via Sparse Linear Regression: Computationally Efficient Encoding and Decoding

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence. It is shown to achieve the optimal distortion-rate function for i.i.d Gaussian sources under the squared-error distortion criterion. For a given rate, the parameters of the design matrix can be varied to trade off distortion performance with encoding complexity. An example of such a trade-off as a function of the block length n is the following. With computational resource (space or time) per source sample of O((n/\log n)^2), for a fixed distortion-level above the Gaussian distortion-rate function, the probability of excess distortion decays exponentially in n. The Sparse Regression Code is robust in the following sense: for any ergodic source, the proposed encoder achieves the optimal distortion-rate function of an i.i.d Gaussian source with the same variance. Simulations show that the encoder has good empirical performance, especially at low and moderate rates.