Matrix analysis
Topics in matrix analysis
Algorithms for the polar decomposition
SIAM Journal on Scientific and Statistical Computing
Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Signal representation using adaptive normalized Gaussian functions
Signal Processing
Sparse Approximate Solutions to Linear Systems
SIAM Journal on Computing
SIAM Review
Optimal wire and transistor sizing for circuits with non-tree topology
ICCAD '97 Proceedings of the 1997 IEEE/ACM international conference on Computer-aided design
Mathematical control theory: deterministic finite dimensional systems (2nd ed.)
Mathematical control theory: deterministic finite dimensional systems (2nd ed.)
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Alternating projection algorithms for linear matrix inequalities problems with rank constraints
Advances in linear matrix inequality methods in control
Distance Matrix Completion by Numerical Optimization
Computational Optimization and Applications
Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones
Mathematics of Operations Research
A Unified Algebric Approach to Control Design
A Unified Algebric Approach to Control Design
An elementary proof of a theorem of Johnson and Lindenstrauss
Random Structures & Algorithms
Database-friendly random projections: Johnson-Lindenstrauss with binary coins
Journal of Computer and System Sciences - Special issu on PODS 2001
Convex Optimization
Local Minima and Convergence in Low-Rank Semidefinite Programming
Mathematical Programming: Series A and B
Learning with matrix factorizations
Learning with matrix factorizations
Fast maximum margin matrix factorization for collaborative prediction
ICML '05 Proceedings of the 22nd international conference on Machine learning
Jordan-Algebraic Approach to Convexity Theorems for Quadratic Mappings
SIAM Journal on Optimization
Survey paper: Structured low-rank approximation and its applications
Automatica (Journal of IFAC)
Interior-Point Method for Nuclear Norm Approximation with Application to System Identification
SIAM Journal on Matrix Analysis and Applications
A Newton-like method for solving rank constrained linear matrix inequalities
Automatica (Journal of IFAC)
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Mirror descent and nonlinear projected subgradient methods for convex optimization
Operations Research Letters
An accelerated gradient method for trace norm minimization
ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
A channel coding perspective of recommendation systems
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 1
Sparse and low-rank matrix decompositions
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Accurate low-rank matrix recovery from a small number of linear measurements
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
The power of convex relaxation: near-optimal matrix completion
IEEE Transactions on Information Theory
Matrix Completion from Noisy Entries
The Journal of Machine Learning Research
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
Kernel-based learning from infinite dimensional 2-way tensors
ICANN'10 Proceedings of the 20th international conference on Artificial neural networks: Part II
Interior-Point Method for Nuclear Norm Approximation with Application to System Identification
SIAM Journal on Matrix Analysis and Applications
Regression on Fixed-Rank Positive Semidefinite Matrices: A Riemannian Approach
The Journal of Machine Learning Research
A Singular Value Thresholding Algorithm for Matrix Completion
SIAM Journal on Optimization
Robust photometric stereo via low-rank matrix completion and recovery
ACCV'10 Proceedings of the 10th Asian conference on Computer vision - Volume Part III
Robust principal component analysis?
Journal of the ACM (JACM)
The minimum-rank gram matrix completion via modified fixed point continuation method
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Integrating low-rank and group-sparse structures for robust multi-task learning
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
Rank aggregation via nuclear norm minimization
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
Alternating Direction Algorithms for $\ell_1$-Problems in Compressive Sensing
SIAM Journal on Scientific Computing
Recovering Low-Rank and Sparse Components of Matrices from Incomplete and Noisy Observations
SIAM Journal on Optimization
Clustering for bioinformatics via matrix optimization
Proceedings of the 2nd ACM Conference on Bioinformatics, Computational Biology and Biomedicine
Exact matrix completion via convex optimization
Communications of the ACM
A Simpler Approach to Matrix Completion
The Journal of Machine Learning Research
Online learning in the embedded manifold of low-rank matrices
The Journal of Machine Learning Research
Metric and kernel learning using a linear transformation
The Journal of Machine Learning Research
On identity testing of tensors, low-rank recovery and compressed sensing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
TILT: Transform Invariant Low-Rank Textures
International Journal of Computer Vision
Linear Analysis of Nonlinear Constraints for Interactive Geometric Modeling
Computer Graphics Forum
Accelerated singular value thresholding for matrix completion
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
Optimal exact least squares rank minimization
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
Low rank modeling of signed networks
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
Accelerated Block-coordinate Relaxation for Regularized Optimization
SIAM Journal on Optimization
Low-rank Matrix Recovery via Iteratively Reweighted Least Squares Minimization
SIAM Journal on Optimization
Restricted strong convexity and weighted matrix completion: optimal bounds with noise
The Journal of Machine Learning Research
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
Spatio-temporal compressive sensing and internet traffic matrices
IEEE/ACM Transactions on Networking (TON)
A fast tri-factorization method for low-rank matrix recovery and completion
Pattern Recognition
Mirror descent for metric learning: a unified approach
ECML PKDD'12 Proceedings of the 2012 European conference on Machine Learning and Knowledge Discovery in Databases - Volume Part I
Background modeling from surveillance video using rank minimization
AICI'12 Proceedings of the 4th international conference on Artificial Intelligence and Computational Intelligence
Accelerated Linearized Bregman Method
Journal of Scientific Computing
Emitter localization using received-strength-signal data
Signal Processing
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
Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials
Computational Optimization and Applications
Fast structure learning in generalized stochastic processes with latent factors
Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining
Robust principal component analysis via capped norms
Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining
Low-rank matrix completion using alternating minimization
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Multisample aCGH Data Analysis via Total Variation and Spectral Regularization
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Sparse and unique nonnegative matrix factorization through data preprocessing
The Journal of Machine Learning Research
Recovering low-rank matrices from corrupted observations via the linear conjugate gradient algorithm
Journal of Computational and Applied Mathematics
Guarantees of augmented trace norm models in tensor recovery
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
Approximation of rank function and its application to the nearest low-rank correlation matrix
Journal of Global Optimization
A reweighted nuclear norm minimization algorithm for low rank matrix recovery
Journal of Computational and Applied Mathematics
Matrix Recipes for Hard Thresholding Methods
Journal of Mathematical Imaging and Vision
Prox-Regularity of Rank Constraint Sets and Implications for Algorithms
Journal of Mathematical Imaging and Vision
A Simple Prior-Free Method for Non-rigid Structure-from-Motion Factorization
International Journal of Computer Vision
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The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.