arXiv subjects

Pablo Pascual Cobo

Publications and source records attributed to Pablo Pascual Cobo.

4 recordsLinked to original sources

arXiv2025

Many-User Multiple Access with Random User Activity: Achievability Bounds and Efficient Schemes

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.

cs.IT↗

arXiv2025

Quantitative Group Testing and Pooled Data in the Linear Regime with Sublinear Tests

In the pooled data problem, 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 a spatially coupled Bernoulli 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 achieves almost-exact recovery with a number of tests sublinear in the total number of items $p$. Although there exist other efficient schemes for noiseless QGT and pooled data that achieve recovery with order-optimal sample complexity ($Θ(\frac{p}{\log p})$ 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 in $p$, 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.

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arXiv2024

Bayes-Optimal Estimation in Generalized Linear Models via Spatial Coupling

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. In this work, 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 in the high-dimensional limit. 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.

cs.IT↗

arXiv2024

Approximate Message Passing with Rigorous Guarantees for Pooled Data and Quantitative Group Testing

In the pooled data problem, 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 of each category within the pool. We study an approximate message passing (AMP) algorithm for estimating the categories and rigorously characterize its performance, in both the noiseless and noisy settings. For the noiseless setting, we show that the AMP algorithm is equivalent to one recently proposed by El Alaoui et al. Our results provide a rigorous version of their performance guarantees, previously obtained via non-rigorous techniques. For the case of pooled data with two categories, known as quantitative group testing (QGT), we use the AMP guarantees to compute precise limiting values of the false positive rate and the false negative rate. Though the pooled data problem and QGT are both instances of estimation in a linear model, existing AMP theory cannot be directly applied since the design matrices are binary valued. The key technical ingredient in our analysis is a rigorous asymptotic characterization of AMP for generalized linear models defined via generalized white noise design matrices. This result, established using a recent universality result of Wang et al., is of independent interest. Our theoretical results are validated by numerical simulations. For comparison, we propose estimators based on convex relaxation and iterative thresholding, without providing theoretical guarantees. The simulations indicate that AMP consistently outperforms these estimators.

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