  This paper is concerned with estimating the column space of an unknown
low-rank matrix $\boldsymbol{A}^{\star}\in\mathbb{R}^{d_{1}\times d_{2}}$,
given noisy and partial observations of its entries. There is no shortage of
scenarios where the observations -- while being too noisy to support faithful
recovery of the entire matrix -- still convey sufficient information to enable
reliable estimation of the column space of interest. This is particularly
evident and crucial for the highly unbalanced case where the column dimension
$d_{2}$ far exceeds the row dimension $d_{1}$, which is the focal point of the
current paper. We investigate an efficient spectral method, which operates upon
the sample Gram matrix with diagonal deletion. While this algorithmic idea has
been studied before, we establish new statistical guarantees for this method in
terms of both $\ell_{2}$ and $\ell_{2,\infty}$ estimation accuracy, which
improve upon prior results if $d_{2}$ is substantially larger than $d_{1}$. To
illustrate the effectiveness of our findings, we derive matching minimax lower
bounds with respect to the noise levels, and develop consequences of our
general theory for three applications of practical importance: (1) tensor
completion from noisy data, (2) covariance estimation / principal component
analysis with missing data, and (3) community recovery in bipartite graphs. Our
theory leads to improved performance guarantees for all three cases.
