# Multi-reference factor analysis: low-rank covariance estimation under unknown translations

## TL;DR

This paper introduces a method for estimating low-rank covariance matrices of signals observed through unknown cyclic shifts and noise, using shift-invariant moments like the power spectrum and trispectrum, with proven consistency and practical effectiveness.

## Contribution

It develops a polynomial-time, statistically consistent procedure for low-rank covariance estimation from translated noisy observations using shift-invariant moments, extending PCA applicability.

## Key findings

- Covariance can be recovered from power spectrum and trispectrum when rank is low.
- The method is statistically consistent and effective even under high noise levels.
- Full-rank covariance matrices cannot be reliably estimated with this approach.

## Abstract

We consider the problem of estimating the covariance matrix of a random signal observed through unknown translations (modeled by cyclic shifts) and corrupted by noise. Solving this problem allows to discover low-rank structures masked by the existence of translations (which act as nuisance parameters), with direct application to Principal Components Analysis (PCA). We assume that the underlying signal is of length $L$ and follows a standard factor model with mean zero and $r$ normally-distributed factors. To recover the covariance matrix in this case, we propose to employ the second- and fourth-order shift-invariant moments of the signal known as the $\textit{power spectrum}$ and the $\textit{trispectrum}$. We prove that they are sufficient for recovering the covariance matrix (under a certain technical condition) when $r<\sqrt{L}$. Correspondingly, we provide a polynomial-time procedure for estimating the covariance matrix from many (translated and noisy) observations, where no explicit knowledge of $r$ is required, and prove the procedure's statistical consistency. While our results establish that covariance estimation is possible from the power spectrum and the trispectrum for low-rank covariance matrices, we prove that this is not the case for full-rank covariance matrices. We conduct numerical experiments that corroborate our theoretical findings, and demonstrate the favorable performance of our algorithms in various settings, including in high levels of noise.