O ct 2 01 9 Subspace Estimation from Unbalanced and Incomplete Data Matrices l 2 Statistical Guarantees
This paper is concerned with estimating the column space of an unknown low rank matrix A R d1xd2 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 d2 far exceeds the row dimension d1, 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. We establish statistical guarantees for this method in terms of both l2 and l2, estimation accuracy, which improve upon prior results if d2 is substantially larger than d1. To illustrate the effectiveness of our findings, we develop consequences of our general theory for three applications of practical importance: (1) tensor completion from noisy data, (2) covariance estimation with missing data, and (3) community recovery in bipartite graphs. Our theory leads to improved performance guarantees for all three cases.