Optimal prediction in the linearly transformed spiked model

Abstract

We consider the linearly transformed spiked model, where the observations $Y_{i}$ are noisy linear transforms of unobserved signals of interest $X_{i}$: \begin{equation*}Y_{i}=A_{i}X_{i}+\varepsilon_{i},\end{equation*} for $i=1,\ldots ,n$. The transform matrices $A_{i}$ are also observed. We model the unobserved signals (or regression coefficients) $X_{i}$ as vectors lying on an unknown low-dimensional space. Given only $Y_{i}$ and $A_{i}$ how should we predict or recover their values? The naive approach of performing regression for each observation separately is inaccurate due to the large noise level. Instead, we develop optimal methods for predicting $X_{i}$ by “borrowing strength” across the different samples. Our linear empirical Bayes methods scale to large datasets and rely on weak moment assumptions. We show that this model has wide-ranging applications in signal processing, deconvolution, cryo-electron microscopy, and missing data with noise. For missing data, we show in simulations that our methods are more robust to noise and to unequal sampling than well-known matrix completion methods.