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. 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 power spectrum and the 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 favourable performance of our algorithms in various settings, including in high levels of noise.
Efficient Full-Rank Spatial Covariance Estimation Using Independent Low-Rank Matrix Analysis for Blind Source Separation