Minimization methods for non-differentiable functions
Minimization methods for non-differentiable functions
Shape and motion from image streams under orthography: a factorization method
International Journal of Computer Vision
Automatica (Journal of IFAC) - Special issue on statistical signal processing and control
N4SID: subspace algorithms for the identification of combined deterministic-stochastic systems
Automatica (Journal of IFAC) - Special issue on statistical signal processing and control
Subspace-based methods for the identification of linear time-invariant systems
Automatica (Journal of IFAC) - Special issue on trends in system identification
Exploiting sparsity in primal-dual interior-point methods for semidefinite programming
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
A Sequential Factorization Method for Recovering Shape and Motion From Image Streams
IEEE Transactions on Pattern Analysis and Machine Intelligence
Self-scaled barriers and interior-point methods for convex programming
Mathematics of Operations Research
System identification (2nd ed.): theory for the user
System identification (2nd ed.): theory for the user
Atomic Decomposition by Basis Pursuit
SIAM Review
On the Nesterov--Todd Direction in Semidefinite Programming
SIAM Journal on Optimization
Optimization Problems over Positive Pseudopolynomial Matrices
SIAM Journal on Matrix Analysis and Applications
Convex Optimization
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Learning with matrix factorizations
Learning with matrix factorizations
SIAM Journal on Optimization
Matrix Methods in Data Mining and Pattern Recognition (Fundamentals of Algorithms)
Matrix Methods in Data Mining and Pattern Recognition (Fundamentals of Algorithms)
Algorithm 875: DSDP5—software for semidefinite programming
ACM Transactions on Mathematical Software (TOMS)
The power of convex relaxation: near-optimal matrix completion
IEEE Transactions on Information Theory
Fixed point and Bregman iterative methods for matrix rank minimization
Mathematical Programming: Series A and B
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Just relax: convex programming methods for identifying sparse signals in noise
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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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.