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Online low-rank subspace clustering by basis dictionary pursuit. arXiv preprint arXiv:1503.08356,
, 2015
"... Abstract Low-Rank Representation (LRR) has been a significant method for segmenting data that are generated from a union of subspaces. It is also known that solving LRR is challenging in terms of time complexity and memory footprint, in that the size of the nuclear norm regularized matrix is n-by-n ..."
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Abstract Low-Rank Representation (LRR) has been a significant method for segmenting data that are generated from a union of subspaces. It is also known that solving LRR is challenging in terms of time complexity and memory footprint, in that the size of the nuclear norm regularized matrix is n-by-n (where n is the number of samples). In this paper, we thereby develop a novel online implementation of LRR that reduces the memory cost from O(n 2 ) to O(pd), with p being the ambient dimension and d being some estimated rank (d < p ≪ n). We also establish the theoretical guarantee that the sequence of solutions produced by our algorithm converges to a stationary point of the expected loss function asymptotically. Extensive experiments on synthetic and realistic datasets further substantiate that our algorithm is fast, robust and memory efficient.
Low-rank tensor completion: a Riemannian manifold preconditioning approach Hiroyuki Kasai
"... Abstract We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that ..."
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Abstract We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms for batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-ofthe-art algorithms across different synthetic and real-world datasets.
