Adaptive Stochastic Gradient Descent on the Grassmannian for Robust Low-Rank Subspace Recovery and Clustering
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In this paper, we present GASG21 (Grassmannian Adaptive Stochastic Gradient for L_2,1 norm minimization), an adaptive stochastic gradient algorithm to robustly recover the low-rank subspace from a large matrix. In the presence of column outliers, we reformulate the batch mode matrix L_2,1 norm minimization with rank constraint problem as a stochastic optimization approach constrained on Grassmann manifold. 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, we choose to adaptively tune the constant step-size by leveraging the consecutive gradients. Furthermore, we demonstrate that with proper initialization, the K-subspaces extension, K-GASG21, can robustly cluster a large number of corrupted data vectors into a union of subspaces. Numerical experiments on synthetic and real data demonstrate the efficiency and accuracy of the proposed algorithms even with heavy column outliers corruption.
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