Optimal prediction in the linearly transformed spiked model

We consider the linearly transformed spiked model, where the observations Y_i are noisy linear transforms of unobserved signals of interest X_i: Y_i = A_iX_i + ε_i, for i = 1, . . ., 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.