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.

本文提出了一种基于Matrix GLM的近似消息传递算法（AMP）来估计混合线性回归等多个广义线性模型的信号，经过状态演化递归的理论分析得出该算法的性能，同时使用状态演化方法在每次迭代中选择AMP的最优去噪函数，数值模拟结果验证了AMP算法对于混合线性回归和max-affine回归等问题的优越表现。

通过近似消息传递实现混合回归