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This paper is concerned with estimating the column space of an unknownlow-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 ofscenarios where the observations -- while being too noisy to support faithfulrecovery of the entire matrix -- still convey sufficient information to enablereliable estimation of the column space of interest. This is particularlyevident 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 thecurrent paper. We investigate an efficient spectral method, which operates uponthe sample Gram matrix with diagonal deletion. While this algorithmic idea hasbeen studied before, we establish new statistical guarantees for this method interms of both $\ell_{2}$ and $\ell_{2,\infty}$ estimation accuracy, whichimprove upon prior results if $d_{2}$ is substantially larger than $d_{1}$. Toillustrate the effectiveness of our findings, we derive matching minimax lowerbounds with respect to the noise levels, and develop consequences of ourgeneral theory for three applications of practical importance: (1) tensorcompletion from noisy data, (2) covariance estimation / principal componentanalysis with missing data, and (3) community recovery in bipartite graphs. Ourtheory leads to improved performance guarantees for all three cases.

Subspace Estimation from Unbalanced and Incomplete Data Matrices: $\ell_{2,\infty}$ Statistical Guarantees