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We study the problem of estimating a large, low-rank matrix corrupted by additive noise of unknown covariance, assuming one has access to additional side information in the form of noise-only measurements. We study the Whiten-Shrink-reColour (WSC) workflow, where a ‘noise covariance whitening’ transformation is applied to the observations, followed by appropriate singular value shrinkage and a ‘noise covariance re-colouring’ transformation. We show that under the mean square error loss, a unique, asymptotically optimal shrinkage nonlinearity exists for the WSC denoising workflow, and calculate it in closed form. To this end, we calculate the asymptotic eigenvector rotation of the random spiked F-matrix ensemble, a result which may be of independent interest. With sufficiently many pure-noise measurements, our optimally tuned WSC denoising workflow outperforms, in mean square error, matrix denoising algorithms based on optimal singular value shrinkage that do not make similar use of noise-only side information; numerical experiments show that our procedure’s relative performance is particularly strong in challenging statistical settings with high dimensionality and large degree of heteroscedasticity. 

Abstract We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in $${\mathbb{R}}^M$$ is uniquely determined when the number $$L$$ of samples per observation is of the order of the square root of the signal’s length ($$L=O(\sqrt{M})$$). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to $$1/\textrm{SNR}^3$$. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled ($L=M$). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.