Subspace Estimation from Incomplete Observations: A High-Dimensional Analysis

Abstract--We present a high-dimensional analysis of three popular algorithms, namely, Oja's method, GROUSE and PETRELS, for subspace estimation from streaming and highly incomplete observations. We show that, with proper time scaling, the time-varying principal angles between the true subspace and its estimates given by the algorithms converge weakly to deterministic processes when the ambient dimension n tends to infinity. Moreover, the limiting processes can be exactly characterized as the unique solutions of certain ordinary differential equations (ODEs). A finite sample bound is also given, showing that the rate of convergence towards such limits is O(1/ n). In addition to providing asymptotically exact predictions of the dynamic performance of the algorithms, our high-dimensional analysis yields several insights, including an asymptotic equivalence between Oja's method and GROUSE, and a precise scaling relationship linking the amount of missing data to the signalto-noise ratio. By analyzing the solutions of the limiting ODEs, we also establish phase transition phenomena associated with the steady-state performance of these techniques. Subspace estimation is a key task in many signal processing applications. Examples include source localization in array processing, system identification, network monitoring, and image sequence analysis, to name a few. The ubiquity of subspace estimation comes from the fact that a low-rank subspace model can conveniently capture the intrinsic, lowdimensional structures of many large datasets.