Subspace Estimation from Incomplete Observations: A Precise High-Dimensional Analysis

The problem of estimating and tracking low-rank subspaces from incomplete observations has received a lot of attention recently in the signal processing and learning communities. Popular algorithms, such as GROUSE [1] and PETRELS [2], are often very effective in practice, but their performance depends on the careful choice of algorithmic parameters. Important questions, such as the global convergence of these algorithms and how the noise level, subsampling ratio, and various other parameters affect the performance, are not fully understood. In this paper, we present a precise analysis of the performance of these algorithms in the asymptotic regime where the ambient dimension tends to infinity. Specifically, we show that the time-varying trajectories of estimation errors converge weakly to a deterministic function of time, which is characterized as the unique solution of a system of ordinary differential equations (ODEs.) Analyzing the limiting ODEs also reveals and characterizes sharp phase transition phenomena associated with these algorithms. Numerical simulations verify the accuracy of our asymptotic predictions, even for moderate signal dimensions.