论文标题
通用线性模型的线性和光谱估计器的最佳组合
Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models
论文作者
论文摘要
我们研究了从具有高斯传感矩阵的广义线性模型获得的测量值的恢复未知信号$ \ boldsymbol x $的问题。两个流行的解决方案基于线性估计器$ \ hat {\ boldsymbol x}^{\ rm l} $和频谱估算器$ \ hat {\ boldsymbol x}^{\ rm s} $。前者是测量矩阵列的数据依赖性线性组合，其分析非常简单。后者是数据依赖性矩阵的主要特征向量，最近的工作线研究了其性能。在本文中，我们展示了如何最佳结合$ \ hat {\ boldsymbol x}^{\ rm l} $和$ \ hat {\ boldsymbol x}^{\ rm s} $。我们分析的核心是$（\ boldsymbol x，\ hat {\ boldsymbol x}^{\ rm l}的关节经验分布的确切表征，\ hat {\ boldsymbol x}^{\ rm s}）在高维极限中。这使我们能够计算$ \ hat {\ boldsymbol x}^{\ rm l} $和$ \ hat {\ boldsymbol x}^{\ rm s} $的贝叶斯最佳组合。当信号的分布是高斯时，贝叶斯最佳组合具有$θ\ hat {\ boldsymbol x}^{\ rm l}+hat {\ boldsymbol x}^{\ rm x}^{\ rm s} $，我们得出最佳组合系数。为了建立$（\ boldsymbol x，\ hat {\ boldsymbol x}^{\ rm l} {\ rm l} {\ hat {\ boldsymbol x}^{\ rm s}）$的限制分布，我们设计并分析了一个大致的消息传递（amp）algorith $ \ hat { l} $和接近$ \ hat {\ boldsymbol x}^{\ rm s} $。数值模拟证明了相对于分别考虑的两种方法的组合的改进。
We study the problem of recovering an unknown signal $\boldsymbol x$ given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator $\hat{\boldsymbol x}^{\rm L}$ and a spectral estimator $\hat{\boldsymbol x}^{\rm s}$. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine $\hat{\boldsymbol x}^{\rm L}$ and $\hat{\boldsymbol x}^{\rm s}$. At the heart of our analysis is the exact characterization of the joint empirical distribution of $(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$ in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of $\hat{\boldsymbol x}^{\rm L}$ and $\hat{\boldsymbol x}^{\rm s}$, given the limiting distribution of the signal $\boldsymbol x$. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form $θ\hat{\boldsymbol x}^{\rm L}+\hat{\boldsymbol x}^{\rm s}$ and we derive the optimal combination coefficient. In order to establish the limiting distribution of $(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$, we design and analyze an Approximate Message Passing (AMP) algorithm whose iterates give $\hat{\boldsymbol x}^{\rm L}$ and approach $\hat{\boldsymbol x}^{\rm s}$. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
