We study the Gaussian multiple access channel with random user activity, in the regime where the number of users is proportional to the code length. The receiver may know some statistics about the number of active users, but does not know the exact number nor the identities of the active users. We derive two achievability bounds on the probabilities of missed detection, false alarm, and active user error, and propose an efficient CDMA-type scheme whose performance can be compared against these bounds. The first bound is a finite-length result based on Gaussian random codebooks and maximum-likelihood decoding. The second is an asymptotic bound, established using spatially coupled Gaussian codebooks and approximate message passing (AMP) decoding. These bounds can be used to compute an achievable tradeoff between the active user density and energy-per-bit, for a fixed user payload and target error rate. The efficient CDMA scheme uses a spatially coupled signature matrix and AMP decoding, and we give rigorous asymptotic guarantees on its error performance. Our analysis provides the first state evolution result for spatially coupled AMP with matrix-valued iterates, which may be of independent interest. Numerical experiments demonstrate the promising error performance of the CDMA scheme for both small and large user payloads, when compared with the two achievability bounds.

In thepooled dataproblem, the goal is to identify the categories associated with a large collection of items via a sequence of pooled tests. Each pooled test reveals the number of items in the pool belonging to each category. A prominent special case is quantitative group testing (QGT), which is the case of pooled data with two categories. We consider these problems in the non-adaptive and linear regime, where the fraction of items in each category is of constant order. We propose a scheme with aspatially coupledBernoulli test matrix and an efficient approximate message passing (AMP) algorithm for recovery. We rigorously characterize its asymptotic performance in both the noiseless and noisy settings, and prove that in the noiseless case, the AMP algorithm achievesalmost-exactrecovery with a number of tests sublinear in the total number of itemsp. Although there exist other efficient schemes for noiseless QGT and pooled data that achieve recovery with order-optimal sample complexity ((Θ(p/logp) tests, there are no guarantees on their performance in the presence of noise, even at low noise-levels. In comparison, our scheme achieves recovery in the noiseless case with a number of tests sublinear inp, and its performance degrades gracefully in the presence of noise. Numerical simulations illustrate the benefits of the spatially coupled scheme at finite dimensions, showing that it outperforms i.i.d. test designs as well as other recovery algorithms based on convex programming.

We study the Gaussian multiple access channel with random user activity, in the regime where the number of users is proportional to the code length. The receiver may know some statistics about the number of active users, but does not know the exact number nor the identities of the active users. We first derive achievability bounds on the error probabilities by analyzing Gaussian random codebooks with maximum-likelihood decoding. We then propose an efficient CDMA-type scheme based on a spatially coupled signature matrix and approximate message passing (AMP) decoding. Rigorous asymptotic guarantees on the error performance of the AMP decoder are derived. A numerical comparison indicates that the asymptotic error guarantees of the spatially coupled scheme are significantly better than those obtained via the finite-length achievability bounds.

We consider the problem of signal estimation in a generalized linear model (GLM). GLMs include many canonical problems in statistical estimation, such as linear regression, phase retrieval, and 1-bit compressed sensing. Recent work has precisely characterized the asymptotic minimum mean-squared error (MMSE) for GLMs with i.i.d. Gaussian sensing matrices. However, in many models there is a significant gap between the MMSE and the performance of the best known feasible estimators. We address this issue by considering GLMs defined via spatially coupled sensing matrices. We propose an efficient approximate message passing (AMP) algorithm for estimation and prove that with a simple choice of spatially coupled design, the MSE of a carefully tuned AMP estimator approaches the asymptotic MMSE as the dimensions of the signal and the observation grow proportionally. To prove the result, we first rigorously characterize the asymptotic performance of AMP for a GLM with a generic spatially coupled design. This characterization is in terms of a deterministic recursion (‘state evolution’) that depends on the parameters defining the spatial coupling. Then, using a simple spatially coupled design and a judicious choice of functions for the AMP algorithm, we analyze the fixed points of the resulting state evolution and show that it achieves the asymptotic MMSE. Numerical results for phase retrieval and rectified linear regression show that spatially coupled designs can yield substantially lower MSE than i.i.d. Gaussian designs at finite dimensions when used with AMP algorithms.

We consider the problem of signal estimation in a generalized linear model (GLM). GLMs cover many canonical problems in statistical estimation including linear regression and phase retrieval. Recent work has precisely characterized the asymptotic minimum mean-squared error (MMSE) for GLMs with i.i.d. Gaussian sensing matrices. However, in many models there is a significant gap between the MMSE and the performance of the best known feasible estimators. In this work we address this gap by considering GLMs defined via spatially coupled sensing matrices. We propose an efficient approximate message passing (AMP) algorithm for estimation and prove that with a simple choice of spatially coupled design, the MSE of a carefully tuned AMP estimator approaches the asymptotic MMSE. Numerical results show that for finite signal dimensions, spatially coupled designs can yield substantially lower MSE than i.i.d. Gaussian designs when used with AMP algorithms.