Rank-one multi-reference factor analysis

In recent years, there is a growing need for processing methods aimed at extracting useful information from large datasets. many cases, the challenge to discover low-dimensional structure in data, often concealed by existence of nuisance parameters and noise. Motivated such challenges, we consider problem estimating signal its scaled, cyclically shifted noisy observations. We focus on particularly challenging regime low signal-to-noise ratio (SNR), where different observations cannot be shift-aligned. show that an accurate estimation possible, derive procedure which proved consistently estimate signal. The asymptotic sample complexity (the number required recover signal) \(1{/}{\text {SNR}}^4\). Additionally, propose experimentally shown improve factor equal signal’s length. Finally, present numerical experiments demonstrate performance our algorithms corroborate theoretical findings.