Adaptive stochastic gradient descent on the Grassmannian for robust low‐rank subspace recovery

In this study, the authors present GASG21 (Grassmannian adaptive stochastic gradient for L 2,1 norm minimisation), an adaptive stochastic gradient algorithm to robustly recover the low‐rank subspace from a large matrix. In the presence of column outliers corruption, the authors reformulate the classical matrix L 2,1 norm minimisation problem as its stochastic programming counterpart. For each observed data vector, the low‐rank subspace S is updated by taking a gradient step along the geodesic of Grassmannian. In order to accelerate the convergence rate of the stochastic gradient method, the authors choose to adaptively tune the constant step‐size by leveraging the consecutive gradients. Numerical experiments on synthetic data and the extended Yale face dataset demonstrate the efficiency and accuracy of the proposed GASG21 algorithm even with heavy column outliers corruption.