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Tight convex relaxations for sparse matrix factorization
, 2014
"... Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of nonzero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and lowrank sparse bilinear regre ..."
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Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of nonzero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and lowrank sparse bilinear regression as potential applications. We compute slow rates and an upper bound on the statistical dimension Amelunxen et al. (2013) of the suggested norm for rank 1 matrices, showing that its statistical dimension is an order of magnitude smaller than the usual `1norm, trace norm and their combinations. Even though our convex formulation is in theory hard and does not lead to provably polynomial time algorithmic schemes, we propose an active set algorithm leveraging the structure of the convex problem to solve it and show promising numerical results.
Learning Structured LowRank Representation via Matrix Factorization
"... Abstract A vast body of recent works in the literature have shown that exploring structures beyond data lowrankness can boost the performance of subspace clustering methods such as LowRank Representation (LRR). It has also been well recognized that the matrix factorization framework might offer mo ..."
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Abstract A vast body of recent works in the literature have shown that exploring structures beyond data lowrankness can boost the performance of subspace clustering methods such as LowRank Representation (LRR). It has also been well recognized that the matrix factorization framework might offer more flexibility on pursuing underlying structures of the data. In this paper, we propose to learn structured LRR by factorizing the nuclear norm regularized matrix, which leads to our proposed nonconvex formulation NLRR. Interestingly, this formulation of NLRR provides a general framework for unifying a variety of popular algorithms including LRR, dictionary learning, robust principal component analysis, sparse subspace clustering, etc. Several variants of NLRR are also proposed, for example, to promote sparsity while preserving lowrankness. We design a practical algorithm for NLRR and its variants, and establish theoretical guarantee for the stability of the solution and the convergence of the algorithm. Perhaps surprisingly, the computational and memory cost of NLRR can be reduced by roughly one order of magnitude compared to the cost of LRR. Experiments on extensive simulations and real datasets confirm the robustness of efficiency of NLRR and the variants.
Online lowrank subspace clustering by basis dictionary pursuit. arXiv preprint arXiv:1503.08356,
, 2015
"... Abstract LowRank 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 nbyn ..."
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Abstract LowRank 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 nbyn (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.
Clustering Consistent Sparse Subspace Clustering
, 2015
"... Subspace clustering is the problem of clustering data points into a union of lowdimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and has been drawing avid attention in machine learning and statistics recently. I ..."
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Subspace clustering is the problem of clustering data points into a union of lowdimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and has been drawing avid attention in machine learning and statistics recently. In particular, a line
Graph Connectivity in Noisy Sparse Subspace Clustering
"... Abstract Subspace clustering is the problem of clustering data points into a union of lowdimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work ..."
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Abstract Subspace clustering is the problem of clustering data points into a union of lowdimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work