Rank minimization for subspace tracking from incomplete data

Abstract

Extracting latent low-dimensional structure from high-dimensional data is of paramount importance in timely inference tasks encountered with `Big Data' analytics. However, increasingly noisy, heterogeneous, and incomplete datasets as well as the need for real-time processing pose major challenges towards achieving this goal. In this context, the fresh look advocated here permeates benefits from rank minimization to track low-dimensional subspaces from incomplete data. Leveraging the low-dimensionality of the subspace sought, a novel estimator is proposed based on an exponentially-weighted least-squares criterion regularized with the nuclear norm. After recasting the non-separable nuclear norm into a form amenable to online optimization, a real-time algorithm is developed and its convergence established under simplifying technical assumptions. The novel subspace tracker can asymptotically offer the well-documented performance guarantees of the batch nuclear-norm regularized estimator. Simulated tests with real Internet data confirm the efficacy of the proposed algorithm in tracking the traffic subspace, and its superior performance relative to state-of-the-art alternatives.