Projected Landweber iteration for matrix completion
Journal of Computational and Applied Mathematics
Spectral Regularization Algorithms for Learning Large Incomplete Matrices
The Journal of Machine Learning Research
ADMiRA: atomic decomposition for minimum rank approximation
IEEE Transactions on Information Theory
A Singular Value Thresholding Algorithm for Matrix Completion
SIAM Journal on Optimization
Robust principal component analysis?
Journal of the ACM (JACM)
International Journal of Sensor Networks
The minimum-rank gram matrix completion via modified fixed point continuation method
Proceedings of the 36th international symposium on Symbolic and algebraic computation
ECML PKDD'11 Proceedings of the 2011 European conference on Machine learning and knowledge discovery in databases - Volume Part III
Trace Norm Regularization: Reformulations, Algorithms, and Multi-Task Learning
SIAM Journal on Optimization
Recovering Low-Rank and Sparse Components of Matrices from Incomplete and Noisy Observations
SIAM Journal on Optimization
Semi-supervised learning with mixed knowledge information
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
Hammerstein system identification using nuclear norm minimization
Automatica (Journal of IFAC)
A fast tri-factorization method for low-rank matrix recovery and completion
Pattern Recognition
Computing real solutions of polynomial systems via low-rank moment matrix completion
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Semi-supervised learning with nuclear norm regularization
Pattern Recognition
Iterative reweighted algorithms for matrix rank minimization
The Journal of Machine Learning Research
Approximation of rank function and its application to the nearest low-rank correlation matrix
Journal of Global Optimization
Matrix Recipes for Hard Thresholding Methods
Journal of Mathematical Imaging and Vision
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The nuclear norm (sum of singular values) of a matrix is often used in convex heuristics for rank minimization problems in control, signal processing, and statistics. Such heuristics can be viewed as extensions of $\ell_1$-norm minimization techniques for cardinality minimization and sparse signal estimation. In this paper we consider the problem of minimizing the nuclear norm of an affine matrix-valued function. This problem can be formulated as a semidefinite program, but the reformulation requires large auxiliary matrix variables, and is expensive to solve by general-purpose interior-point solvers. We show that problem structure in the semidefinite programming formulation can be exploited to develop more efficient implementations of interior-point methods. In the fast implementation, the cost per iteration is reduced to a quartic function of the problem dimensions and is comparable to the cost of solving the approximation problem in the Frobenius norm. In the second part of the paper, the nuclear norm approximation algorithm is applied to system identification. A variant of a simple subspace algorithm is presented in which low-rank matrix approximations are computed via nuclear norm minimization instead of the singular value decomposition. This has the important advantage of preserving linear matrix structure in the low-rank approximation. The method is shown to perform well on publicly available benchmark data.