arXiv 2008.033262020-08stat.MLcs.ITcs.LGmath.ITmath.STstat.TH

广义线性模型线性估计和谱估计的最优组合

Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models

Institute of Science and Technology (IST) Austria（奥地利科学技术研究所）；University of British Columbia (UBC)（不列颠哥伦比亚大学）；University of Cambridge（剑桥大学）

Marco Mondelli, Christos Thrampoulidis, Ramji Venkataramanan

备注：: 49 pages, 6 figures

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摘要：

我们研究的问题，恢复一个未知的信号$\boldsymbol x$给定的测量得到的广义线性模型与高斯传感矩阵。两种流行的解决方案基于线性估计器$\hat{\boldsymbol x}^{\rm L}$和谱估计器$\hat{\boldsymbol x}^{\rm s}$。前者是测量矩阵列的数据依赖线性组合，其分析相当简单。后者是数据依赖矩阵的主特征向量，最近的一项工作研究了它的性能。在本文中，我们展示了如何最佳地组合$\hat{\boldsymbol x}^{\rm L}$和$\hat{\boldsymbol x}^{\rm s}$。在我们的分析的核心是联合经验分布的$(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$在高维极限的准确表征。这允许我们计算$\hat{\boldsymbol x}^{\rm L}$和$\hat{\boldsymbol x}^{\rm s}$的贝叶斯最优组合，给定信号$\boldsymbol x$的极限分布。当信号的分布为高斯分布时，贝叶斯最优组合具有$θ\hat{\boldsymbol x}^{\rm L}+\hat{\boldsymbol x}^{\rm s}$的形式，并且我们导出了最优组合系数。为了建立$(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$的极限分布，我们设计并分析了一个近似消息传递（AMP）算法，它的迭代给出了$\hat{\boldsymbol x}^{\rm L}$并逼近了$\hat{\boldsymbol x}^{\rm s}$ .数值模拟表明，相对于单独考虑的两种方法，所提出的组合的改进。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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