线性转换加标模型中的最优预测
Optimal prediction in the linearly transformed spiked model
摘要：
我们考虑线性变换的尖峰模型，其中观测$Y_i$是未观测到的感兴趣信号的噪声线性变换$X_i$：\begin{align*} Y_i = A_i X_i + \varepsilon_i, \end{align*} for $i=1,\ldots,n$。还观察到变换矩阵$A_i$。我们将$X_i$建模为位于未知低维空间上的随机向量。我们应该如何预测未观察到的信号（回归系数）$X_i$？由于噪声很大，单独对每个观测值执行回归的简单方法是不准确的。相反，我们开发了最佳的线性经验贝叶斯方法预测$X_i$的“借款强度”在不同的样本。我们的方法适用于大型数据集，并依赖于弱矩假设。分析是基于随机矩阵理论。我们讨论的应用程序，信号处理，反卷积，低温电子显微镜，和丢失的数据在高噪声制度。对于缺失的数据，我们在模拟中表明，我们的方法是更快，更强大的噪声和不平等的采样比著名的矩阵完成方法。
英文摘要：
We consider the linearly transformed spiked model, where observations $Y_i$ are noisy linear transforms of unobserved signals of interest $X_i$: \begin{align*} Y_i = A_i X_i + \varepsilon_i, \end{align*} for $i=1,\ldots,n$. The transform matrices $A_i$ are also observed. We model $X_i$ as random vectors lying on an unknown low-dimensional space. How should we predict the unobserved signals (regression coefficients) $X_i$? The naive approach of performing regression for each observation separately is inaccurate due to the large noise. Instead, we develop optimal linear empirical Bayes methods for predicting $X_i$ by "borrowing strength" across the different samples. Our methods are applicable to large datasets and rely on weak moment assumptions. The analysis is based on random matrix theory. We discuss applications to signal processing, deconvolution, cryo-electron microscopy, and missing data in the high-noise regime. For missing data, we show in simulations that our methods are faster, more robust to noise and to unequal sampling than well-known matrix completion methods.
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