We consider the problem of localizing change points in a generalized linear model (GLM), a model that covers many widely studied problems in statistical learning including linear, logistic, and rectified linear regression. We propose a novel and computationally efficient Approximate Message Passing (AMP) algorithm for estimating both the signals and the change point locations, and rigorously characterize its performance in the high-dimensional limit where the number of parameters $p$ is proportional to the number of samples $n$. This characterization is in terms of a state evolution recursion, which allows us to precisely compute performance measures such as the asymptotic Hausdorff error of our change point estimates, and allows us to tailor the algorithm to take advantage of any prior structural information on the signals and change points. Moreover, we show how our AMP iterates can be used to efficiently compute a Bayesian posterior distribution over the change point locations in the high-dimensional limit. We validate our theory via numerical experiments, and demonstrate the favorable performance of our estimators on both synthetic and real data in the settings of linear, logistic, and rectified linear regression.

翻译：我们考虑广义线性模型（GLM）中的变点定位问题，该模型涵盖了统计学习中广泛研究的诸多问题，包括线性回归、逻辑回归和修正线性回归。我们提出了一种新颖且计算高效的近似消息传递（AMP）算法，用于同时估计信号和变点位置，并在参数数量 $p$ 与样本数量 $n$ 成比例的高维极限下严格刻画了其性能。该刻画通过状态演化递归方程实现，使我们能够精确计算性能指标（例如变点估计的渐近豪斯多夫误差），并允许我们调整算法以利用信号和变点的任何先验结构信息。此外，我们展示了如何利用AMP迭代在高维极限下高效计算变点位置的贝叶斯后验分布。我们通过数值实验验证了理论结果，并在线性回归、逻辑回归和修正线性回归场景下，通过合成数据与真实数据证明了所提估计器的优越性能。