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In this paper, we present GASG21 (Grassmannian Adaptive Stochastic Gradientfor $L_{2,1}$ norm minimization), an adaptive stochastic gradient algorithm torobustly recover the low-rank subspace from a large matrix. In the presence ofcolumn outliers, we reformulate the batch mode matrix $L_{2,1}$ normminimization with rank constraint problem as a stochastic optimization approachconstrained on Grassmann manifold. For each observed data vector, the low-ranksubspace $\mathcal{S}$ is updated by taking a gradient step along the geodesicof Grassmannian. In order to accelerate the convergence rate of the stochasticgradient method, we choose to adaptively tune the constant step-size byleveraging the consecutive gradients. Furthermore, we demonstrate that withproper initialization, the K-subspaces extension, K-GASG21, can robustlycluster a large number of corrupted data vectors into a union of subspaces.Numerical experiments on synthetic and real data demonstrate the efficiency andaccuracy of the proposed algorithms even with heavy column outliers corruption.

Adaptive Stochastic Gradient Descent on the Grassmannian for Robust Low-Rank Subspace Recovery and Clustering