Generalized approximate message passing (GAMP) is a computationally efficient algorithm for estimating an unknown signal $w_0\in\mathbb{R}^N$ from a random linear measurement $y= Xw_0 + \epsilon\in\mathbb{R}^M$, where $X\in\mathbb{R}^{M\times N}$ is a known measurement matrix and $\epsilon$ is the noise vector. The salient feature of GAMP is that it can provide an unbiased estimator $\hat{r}^{\rm G}\sim\mathcal{N}(w_0, \hat{s}^2I_N)$, which can be used for various hypothesis-testing methods. In this study, we consider the bootstrap average of an unbiased estimator of GAMP for the elastic net. By numerically analyzing the state evolution of \emph{approximate message passing with resampling}, which has been proposed for computing bootstrap statistics of the elastic net estimator, we investigate when the bootstrap averaging reduces the variance of the unbiased estimator and the effect of optimizing the size of each bootstrap sample and hyperparameter of the elastic net regularization in the asymptotic setting $M, N\to\infty, M/N\to\alpha\in(0,\infty)$. The results indicate that bootstrap averaging effectively reduces the variance of the unbiased estimator when the actual data generation process is inconsistent with the sparsity assumption of the regularization and the sample size is small. Furthermore, we find that when $w_0$ is less sparse, and the data size is small, the system undergoes a phase transition. The phase transition indicates the existence of the region where the ensemble average of unbiased estimators of GAMP for the elastic net norm minimization problem yields the unbiased estimator with the minimum variance.

翻译：广义近似消息传递（GAMP）是一种计算高效的算法，用于从随机线性测量 $y= Xw_0 + \epsilon\in\mathbb{R}^M$ 中估计未知信号 $w_0\in\mathbb{R}^N$，其中 $X\in\mathbb{R}^{M\times N}$ 是已知测量矩阵，$\epsilon$ 是噪声向量。GAMP的显著特点是能提供无偏估计量 $\hat{r}^{\rm G}\sim\mathcal{N}(w_0, \hat{s}^2I_N)$，该估计量可用于多种假设检验方法。在本研究中，我们考虑弹性网GAMP无偏估计量的Bootstrap平均。通过数值分析针对弹性网估计量Bootstrap统计量提出的重采样近似消息传递的状态演化，我们研究了在渐近设定 $M, N\to\infty, M/N\to\alpha\in(0,\infty)$ 下，Bootstrap平均何时能降低无偏估计量的方差，以及优化每个Bootstrap样本大小和弹性网正则化超参数的效果。结果表明：当实际数据生成过程与正则化的稀疏性假设不一致且样本量较小时，Bootstrap平均能有效降低无偏估计量的方差。此外，我们发现当 $w_0$ 稀疏度较低且数据量较小时，系统会发生相变。该相变表明存在一个区域，在该区域内弹性网范数最小化问题的GAMP无偏估计量的系综平均能产生具有最小方差的无偏估计量。