Optimal prediction in the linearly transformed spiked model

TL;DR

This paper develops optimal linear empirical Bayes methods for predicting unobserved signals in a linearly transformed spiked model, leveraging random matrix theory to improve accuracy in high-noise, large-data scenarios.

Contribution

It introduces novel empirical Bayes prediction techniques for the linearly transformed spiked model, applicable to large datasets with weak assumptions, outperforming existing methods in noisy, missing data contexts.

Findings

Methods are effective for large datasets with high noise.

Predictions outperform traditional matrix completion methods.

Approach is robust to noise and unequal sampling.

Abstract

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.