  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.
