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We consider the problem of estimating the factors of a low-rank $n \times d$matrix, when this is corrupted by additive Gaussian noise. A special example ofour setting corresponds to clustering mixtures of Gaussians with equal (known)covariances. Simple spectral methods do not take into account the distributionof the entries of these factors and are therefore often suboptimal. Here, wecharacterize the asymptotics of the minimum estimation error under theassumption that the distribution of the entries is known to the statistician.  Our results apply to the high-dimensional regime $n, d \to \infty$ and $d / n\to \infty$ (or $d / n \to 0$) and generalize earlier work that focused on theproportional asymptotics $n, d \to \infty$, $d / n \to \delta \in (0, \infty)$.We outline an interesting signal strength regime in which $d / n \to \infty$and partial recovery is possible for the left singular vectors while impossiblefor the right singular vectors.  We illustrate the general theory by deriving consequences for Gaussianmixture clustering and carrying out a numerical study on genomics data.

Fundamental Limits of Low-Rank Matrix Estimation with Diverging Aspect Ratios