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 result is a rigorous analysis 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. Our simulations indicate that AMP outperforms the convex programming estimator for a range of QGT scenarios, but the convex program performs better for pooled data with three categories.

Mixed Regression via Approximate Message Passing

We study the problem of regression in a generalized linear model (GLM) with multiple signals and latent variables. This model, which we call a matrix GLM, covers many widely studied problems in statistical learning, including mixed linear regression, max-affine regression, and mixture-of-experts. In mixed linear regression, each observation comes from one of $L$ signal vectors (regressors), but we do not know which one; in max-affine regression, each observation comes from the maximum of $L$ affine functions, each defined via a different signal vector. The goal in all these problems is to estimate the signals, and possibly some of the latent variables, from the observations. We propose a novel approximate message passing (AMP) algorithm for estimation in a matrix GLM and rigorously characterize its performance in the high-dimensional limit. This characterization is in terms of a state evolution recursion, which allows us to precisely compute performance measures such as the asymptotic mean-squared error. The state evolution characterization can be used to tailor the AMP algorithm to take advantage of any structural information known about the signals. Using state evolution, we derive an optimal choice of AMP `denoising' functions that minimizes the estimation error in each iteration. The theoretical results are validated by numerical simulations for mixed linear regression, max-affine regression, and mixture-of-experts. For max-affine regression, we propose an algorithm that combines AMP with expectation-maximization to estimate intercepts of the model along with the signals. The numerical results show that AMP significantly outperforms other estimators for mixed linear regression and max-affine regression in most parameter regimes.