Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise, a setting that is also known as “spiked model” or “deformed random matrix.” If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic accuracy of Bayes-optimal estimators in the high-dimensional limit. In this paper, we present a practical algorithm that can achieve Bayes-optimal accuracy above the spectral threshold. A bold conjecture from statistical physics posits that no polynomial-time algorithm achieves optimal error below the same threshold (unless the best estimator is trivial). Our approach uses Approximate Message Passing (AMP) in conjunction with a spectral initialization. AMP algorithms have proved successful in a variety of statistical estimation tasks, and are amenable to exact asymptotic analysis via state evolution. Unfortunately, state evolution is uninformative when the algorithm is initialized near an unstable fixed point, as often happens in low-rank matrix estimation problems. We develop a new analysis of AMP that allows for spectral initializations, and builds on a decoupling between the outlier eigenvectors and the bulk in the spiked random matrix model. Our main theorem is general and applies beyond matrix estimation. However, we use it to derive detailed predictions for the problem of estimating a rank-one matrix in noise. Special cases of this problem are closely related—via universality arguments—to the network community detection problem for two asymmetric communities. For general rank-one models, we show that AMP can be used to construct confidence intervals and control false discovery rate. We provide illustrations of the general methodology by considering the cases of sparse low-rank matrices and of block-constant low-rank matrices with symmetric blocks (we refer to the latter as to the “Gaussian block model”).

中文翻译：

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通过近似消息传递来估计低秩矩阵

考虑当高斯噪声扰动低秩矩阵的项时估计低秩矩阵的问题，该设置也称为“尖峰模型”或“变形随机矩阵”。如果知道尖峰项的经验分布，则利用此知识的最佳估计量将明显优于简单的光谱方法。最近的工作描述了在高维极限中贝叶斯最优估计量的渐近精度。在本文中，我们提出了一种实用的算法，可以在光谱阈值以上实现贝叶斯最佳精度。统计物理学的一个大胆猜测认为，没有多项式时间算法可以在同一阈值以下实现最佳误差（除非最佳估计量是微不足道的）。我们的方法将近似消息传递（AMP）与频谱初始化结合使用。AMP算法已被证明在各种统计估计任务中都是成功的，并且可以通过状态演化进行精确的渐近分析。不幸的是，当算法在不稳定的固定点附近初始化时，状态演化是无用的，这在低秩矩阵估计问题中经常发生。我们开发了一种新的AMP分析，可以进行光谱初始化，并建立在异常特征向量与加标随机矩阵模型中的主体之间的去耦基础上。我们的主要定理是一般性定理，不仅仅适用于矩阵估计。但是，我们使用它来得出有关估计噪声中的秩矩阵的问题的详细预测。该问题的特殊情况（通过通用性论点）与两个不对称社区的网络社区检测问题密切相关。对于一般的排名第一的模型，我们表明AMP可用于构建置信区间并控制错误发现率。我们通过考虑稀疏低秩矩阵和具有对称块的块常数低秩矩阵的情况来提供一般方法的说明（我们将后者称为“高斯块模型”）。