The Annals of Statistics, ISSN 0090-5364, 8/2012, Volume 40, Issue 4, pp. 2195 - 2238

This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower-dimensional planes. As is common in...

Outliers | Polytopes | Computer vision | Determinism | Mathematical theorems | Optimal solutions | Eigenvalues | Computer pattern recognition | Laplacians | Computer conferencing | geometric functional analysis | INFORMATION | properties of convex bodies | Subspace clustering | STATISTICS & PROBABILITY | COMPRESSION | outlier detection | l minimization | MOTION SEGMENTATION | duality in linear programming | concentration of measure | MODELS | IMAGES | BODIES | spectral clustering | Studies | Cluster analysis | Algorithms | 62-07 | ell_{1} minimization

Outliers | Polytopes | Computer vision | Determinism | Mathematical theorems | Optimal solutions | Eigenvalues | Computer pattern recognition | Laplacians | Computer conferencing | geometric functional analysis | INFORMATION | properties of convex bodies | Subspace clustering | STATISTICS & PROBABILITY | COMPRESSION | outlier detection | l minimization | MOTION SEGMENTATION | duality in linear programming | concentration of measure | MODELS | IMAGES | BODIES | spectral clustering | Studies | Cluster analysis | Algorithms | 62-07 | ell_{1} minimization

Journal Article

Journal of the ACM (JACM), ISSN 0004-5411, 05/2011, Volume 58, Issue 3, pp. 1 - 37

This article is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we...

low-rank matrices | nuclear-norm minimization | sparsity | 1 -norm minimization | duality | robustness vis-a-vis outliers | video surveillance | Principal components | Sparsity | Nuclear-norm minimization | Robustness vis-a-vis outliers | Video surveillance | Duality | Low-rank matrices | norm minimization | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | APPROXIMATION | Theory | COMPUTER SCIENCE, INFORMATION SYSTEMS | MATRIX COMPLETION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | l-norm minimization | Algorithms | COMPUTER SCIENCE, THEORY & METHODS | Mathematical problems | Studies | Optimization algorithms | Principal components analysis | Matrix | Shadows | Face recognition | Surveillance | Methodology | Norms | Optimization | Principal component analysis | Image detection

low-rank matrices | nuclear-norm minimization | sparsity | 1 -norm minimization | duality | robustness vis-a-vis outliers | video surveillance | Principal components | Sparsity | Nuclear-norm minimization | Robustness vis-a-vis outliers | Video surveillance | Duality | Low-rank matrices | norm minimization | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | APPROXIMATION | Theory | COMPUTER SCIENCE, INFORMATION SYSTEMS | MATRIX COMPLETION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | l-norm minimization | Algorithms | COMPUTER SCIENCE, THEORY & METHODS | Mathematical problems | Studies | Optimization algorithms | Principal components analysis | Matrix | Shadows | Face recognition | Surveillance | Methodology | Norms | Optimization | Principal component analysis | Image detection

Journal Article

Foundations of Computational Mathematics, ISSN 1615-3375, 12/2009, Volume 9, Issue 6, pp. 717 - 772

We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries...

Nuclear norm minimization | Random matrices | Decoupling | Noncommutative Khintchine inequality | Convex optimization | Duality in optimization | Compressed sensing | Low-rank matrices | Matrix completion | MATHEMATICS, APPLIED | INEQUALITIES | U-STATISTICS | MATHEMATICS | COMPUTER SCIENCE, THEORY & METHODS

Nuclear norm minimization | Random matrices | Decoupling | Noncommutative Khintchine inequality | Convex optimization | Duality in optimization | Compressed sensing | Low-rank matrices | Matrix completion | MATHEMATICS, APPLIED | INEQUALITIES | U-STATISTICS | MATHEMATICS | COMPUTER SCIENCE, THEORY & METHODS

Journal Article

The Annals of Statistics, ISSN 0090-5364, 10/2015, Volume 43, Issue 5, pp. 2055 - 2085

In many fields of science, we observe a response variable together with a large number of potential explanatory variables, and would like to be able to...

Sequential hypothesis testing | Martingale theory | False discovery rate (FDR) | Lasso | Permutation methods | Variable selection | permutation methods | CONFIDENCE-INTERVALS | MODEL SELECTION | martingale theory | STATISTICS & PROBABILITY | false discovery rate (FDR) | sequential hypothesis testing | Studies | Variables | Hypothesis testing | Measurement errors | Statistical methods | Mathematical models | 62J05 | 62F03

Sequential hypothesis testing | Martingale theory | False discovery rate (FDR) | Lasso | Permutation methods | Variable selection | permutation methods | CONFIDENCE-INTERVALS | MODEL SELECTION | martingale theory | STATISTICS & PROBABILITY | false discovery rate (FDR) | sequential hypothesis testing | Studies | Variables | Hypothesis testing | Measurement errors | Statistical methods | Mathematical models | 62J05 | 62F03

Journal Article

The Annals of Statistics, ISSN 0090-5364, 4/2014, Volume 42, Issue 2, pp. 669 - 699

Subspace clustering refers to the task of finding a multi-subspace representation that best fits a collection of points taken from a high-dimensional space....

Computer vision | Error rates | Algorithms | High dimensional spaces | Heuristics | Signal noise | Principal components analysis | Matrices | International conferences | Mathematical vectors | True and false discoveries | Multiple hypothesis testing | LASSO | Subspace clustering | Geometric functional analysis | Spectral clustering | ℓ1 minimization | Dantzig selector | Nonasymptotic random matrix theory | geometric functional analysis | multiple hypothesis testing | ALGORITHM | nonasymptotic random matrix theory | STATISTICS & PROBABILITY | HIGH-DIMENSIONAL REGRESSION | l minimization | GRAPHS | RECOVERY | SEGMENTATION | true and false discoveries | SWITCHED ARX SYSTEMS | spectral clustering | Geometry | Error correction & detection | Topology | Pattern recognition | Clustering | Effectiveness studies | 62-07 | ell_{1} minimization

Computer vision | Error rates | Algorithms | High dimensional spaces | Heuristics | Signal noise | Principal components analysis | Matrices | International conferences | Mathematical vectors | True and false discoveries | Multiple hypothesis testing | LASSO | Subspace clustering | Geometric functional analysis | Spectral clustering | ℓ1 minimization | Dantzig selector | Nonasymptotic random matrix theory | geometric functional analysis | multiple hypothesis testing | ALGORITHM | nonasymptotic random matrix theory | STATISTICS & PROBABILITY | HIGH-DIMENSIONAL REGRESSION | l minimization | GRAPHS | RECOVERY | SEGMENTATION | true and false discoveries | SWITCHED ARX SYSTEMS | spectral clustering | Geometry | Error correction & detection | Topology | Pattern recognition | Clustering | Effectiveness studies | 62-07 | ell_{1} minimization

Journal Article

Foundations of Computational Mathematics, ISSN 1615-3375, 06/2015, Volume 15, Issue 3, pp. 715 - 732

In this paper we introduce a simple heuristic adaptive restart technique that can dramatically improve the convergence rate of accelerated gradient schemes....

Convex optimization | First order methods | Accelerated gradient schemes | MATHEMATICS | MATHEMATICS, APPLIED | LINEAR INVERSE PROBLEMS | SHRINKAGE | THRESHOLDING ALGORITHM | COMPUTER SCIENCE, THEORY & METHODS | Analysis | Convergence (Mathematics) | Iterative methods (Mathematics) | Optimization algorithms | Heuristics | Computational mathematics

Convex optimization | First order methods | Accelerated gradient schemes | MATHEMATICS | MATHEMATICS, APPLIED | LINEAR INVERSE PROBLEMS | SHRINKAGE | THRESHOLDING ALGORITHM | COMPUTER SCIENCE, THEORY & METHODS | Analysis | Convergence (Mathematics) | Iterative methods (Mathematics) | Optimization algorithms | Heuristics | Computational mathematics

Journal Article

SIAM Journal on Imaging Sciences, ISSN 1936-4954, 2011, Volume 4, Issue 1, pp. 1 - 39

Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature...

Smooth approximations of nonsmooth functions | Duality in convex optimization | Minimization | Nesterov's method | Total-variation minimization | Compressed sensing | Continuation methods | MATHEMATICS, APPLIED | compressed sensing | GRADIENT METHODS | IMAGE-RESTORATION | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | L-NORM MINIMIZATION | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | continuation methods | BREGMAN ITERATION | LEAST-SQUARES | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | l minimization | COMPUTER SCIENCE, SOFTWARE ENGINEERING | smooth approximations of nonsmooth functions | total-variation minimization | LINEAR INVERSE PROBLEMS | duality in convex optimization | TOTAL VARIATION MINIMIZATION | SUBSPACE OPTIMIZATION | Studies | Algorithms | Sparsity | Convex analysis | Mathematics - Optimization and Control

Smooth approximations of nonsmooth functions | Duality in convex optimization | Minimization | Nesterov's method | Total-variation minimization | Compressed sensing | Continuation methods | MATHEMATICS, APPLIED | compressed sensing | GRADIENT METHODS | IMAGE-RESTORATION | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | L-NORM MINIMIZATION | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | continuation methods | BREGMAN ITERATION | LEAST-SQUARES | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | l minimization | COMPUTER SCIENCE, SOFTWARE ENGINEERING | smooth approximations of nonsmooth functions | total-variation minimization | LINEAR INVERSE PROBLEMS | duality in convex optimization | TOTAL VARIATION MINIMIZATION | SUBSPACE OPTIMIZATION | Studies | Algorithms | Sparsity | Convex analysis | Mathematics - Optimization and Control

Journal Article

Communications on Pure and Applied Mathematics, ISSN 0010-3640, 06/2014, Volume 67, Issue 6, pp. 906 - 956

This paper develops a mathematical theory of super‐resolution. Broadly speaking, super‐resolution is the problem of recovering the fine details of an...

SUPERRESOLUTION | SPARSITY | MATHEMATICS | MATHEMATICS, APPLIED | UNCERTAINTY PRINCIPLES | RECONSTRUCTION | ALGORITHM | SIGNALS | INNOVATION | FINITE RATE | Mathematical problems | Theory | Optimization

SUPERRESOLUTION | SPARSITY | MATHEMATICS | MATHEMATICS, APPLIED | UNCERTAINTY PRINCIPLES | RECONSTRUCTION | ALGORITHM | SIGNALS | INNOVATION | FINITE RATE | Mathematical problems | Theory | Optimization

Journal Article

Communications on Pure and Applied Mathematics, ISSN 0010-3640, 08/2013, Volume 66, Issue 8, pp. 1241 - 1274

Suppose we wish to recover a signal \input amssym $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} {\bi x} \in {\Bbb C}^n$ from m intensity measurements of the form...

MATHEMATICS | MATHEMATICS, APPLIED | RETRIEVAL | Mathematical problems | Measurement | Probability distribution | Mathematical programming

MATHEMATICS | MATHEMATICS, APPLIED | RETRIEVAL | Mathematical problems | Measurement | Probability distribution | Mathematical programming

Journal Article

Magnetic Resonance in Medicine, ISSN 0740-3194, 03/2015, Volume 73, Issue 3, pp. 1125 - 1136

PurposeTo apply the low-rank plus sparse (L+S) matrix decomposition model to reconstruct undersampled dynamic MRI as a superposition of background and dynamic...

low‐rank matrix completion | sparsity | compressed sensing | dynamic MRI | Sparsity | Dynamic MRI | Low-rank matrix completion | Compressed sensing | IMAGE-RECONSTRUCTION | TRAJECTORIES | THRESHOLDING ALGORITHM | COMBINATION | CONSTRAINT | INTERPOLATION | low-rank matrix completion | LINEAR INVERSE PROBLEMS | ARBITRARY K-SPACE | RADIOLOGY, NUCLEAR MEDICINE & MEDICAL IMAGING | Data Interpretation, Statistical | Reproducibility of Results | Sample Size | Algorithms | Magnetic Resonance Imaging, Cine - methods | Numerical Analysis, Computer-Assisted | Humans | Image Interpretation, Computer-Assisted - methods | Sensitivity and Specificity | Signal Processing, Computer-Assisted | Image Enhancement - methods | Subtraction Technique | Index Medicus

low‐rank matrix completion | sparsity | compressed sensing | dynamic MRI | Sparsity | Dynamic MRI | Low-rank matrix completion | Compressed sensing | IMAGE-RECONSTRUCTION | TRAJECTORIES | THRESHOLDING ALGORITHM | COMBINATION | CONSTRAINT | INTERPOLATION | low-rank matrix completion | LINEAR INVERSE PROBLEMS | ARBITRARY K-SPACE | RADIOLOGY, NUCLEAR MEDICINE & MEDICAL IMAGING | Data Interpretation, Statistical | Reproducibility of Results | Sample Size | Algorithms | Magnetic Resonance Imaging, Cine - methods | Numerical Analysis, Computer-Assisted | Humans | Image Interpretation, Computer-Assisted - methods | Sensitivity and Specificity | Signal Processing, Computer-Assisted | Image Enhancement - methods | Subtraction Technique | Index Medicus

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 04/2015, Volume 61, Issue 4, pp. 1985 - 2007

We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x ∈ ℂ n about which we...

non-convex optimization | Fourier transforms | Accuracy | Diffraction | Computational modeling | Wirtinger derivatives | convergence to global optimum | phase retrieval | Vectors | Optimization | Convergence | Non-convex optimization | X-RAY CRYSTALLOGRAPHY | RECONSTRUCTION | COMPUTER SCIENCE, INFORMATION SYSTEMS | MATRIX COMPLETION | MULTIDIMENSIONAL SEQUENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | MAGNITUDE | FOURIER-TRANSFORM | ALLOW | Measurement | Usage | Frequency modulation | Innovations | Signal processing | Convex functions | Mathematical optimization | Iterative methods (Mathematics) | Coding theory | Information retrieval | Codes | Algorithms | Information theory

non-convex optimization | Fourier transforms | Accuracy | Diffraction | Computational modeling | Wirtinger derivatives | convergence to global optimum | phase retrieval | Vectors | Optimization | Convergence | Non-convex optimization | X-RAY CRYSTALLOGRAPHY | RECONSTRUCTION | COMPUTER SCIENCE, INFORMATION SYSTEMS | MATRIX COMPLETION | MULTIDIMENSIONAL SEQUENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | MAGNITUDE | FOURIER-TRANSFORM | ALLOW | Measurement | Usage | Frequency modulation | Innovations | Signal processing | Convex functions | Mathematical optimization | Iterative methods (Mathematics) | Coding theory | Information retrieval | Codes | Algorithms | Information theory

Journal Article

Inverse Problems, ISSN 0266-5611, 06/2007, Volume 23, Issue 3, pp. 969 - 985

We consider the problem of reconstructing a sparse signal x(0) is an element of R-n from a limited number of linear measurements. Given m randomly selected...

MATHEMATICS, APPLIED | EMPIRICAL PROCESSES | BASES | REPRESENTATIONS | SIGNAL RECOVERY | CONCENTRATION INEQUALITIES | MAXIMA | PHYSICS, MATHEMATICAL | ROBUST UNCERTAINTY PRINCIPLES

MATHEMATICS, APPLIED | EMPIRICAL PROCESSES | BASES | REPRESENTATIONS | SIGNAL RECOVERY | CONCENTRATION INEQUALITIES | MAXIMA | PHYSICS, MATHEMATICAL | ROBUST UNCERTAINTY PRINCIPLES

Journal Article

Comptes rendus - Mathématique, ISSN 1631-073X, 2008, Volume 346, Issue 9, pp. 589 - 592

It is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with...

MATHEMATICS

MATHEMATICS

Journal Article

The Annals of Statistics, ISSN 0090-5364, 2/2011, Volume 39, Issue 1, pp. 278 - 304

We consider the problem of detecting whether or not, in a given sensor network, there is a cluster of sensors which exhibit an "unusual behavior." Formally,...

Minimax | Surveillance | Disease models | Spacetime | Disease outbreaks | Geometric shapes | Euclidean space | Sensors | Statistics | Cylinders | Generalized likelihood ratio test | Scan statistic | Bayesian detection | Disease outbreak detection | Sensor networks | Cellular automata | Detecting a cluster of nodes in a network | Minimax detection | Richardson's model | DEFORMABLE TEMPLATE | DISEASE SURVEILLANCE | SIGNALS | CLASSIFICATION | STATISTICS & PROBABILITY | sensor networks | scan statistic | cellular automata | HIGHER CRITICISM | RANDOM GROWTH | disease outbreak detection | SCAN STATISTICS | TRACKING | minimax detection | MAXIMA | generalized likelihood ratio test | Studies | Cluster analysis | Mathematical models | Richardson’s model | 62C20 | 62G10 | 82B20

Minimax | Surveillance | Disease models | Spacetime | Disease outbreaks | Geometric shapes | Euclidean space | Sensors | Statistics | Cylinders | Generalized likelihood ratio test | Scan statistic | Bayesian detection | Disease outbreak detection | Sensor networks | Cellular automata | Detecting a cluster of nodes in a network | Minimax detection | Richardson's model | DEFORMABLE TEMPLATE | DISEASE SURVEILLANCE | SIGNALS | CLASSIFICATION | STATISTICS & PROBABILITY | sensor networks | scan statistic | cellular automata | HIGHER CRITICISM | RANDOM GROWTH | disease outbreak detection | SCAN STATISTICS | TRACKING | minimax detection | MAXIMA | generalized likelihood ratio test | Studies | Cluster analysis | Mathematical models | Richardson’s model | 62C20 | 62G10 | 82B20

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 04/2011, Volume 57, Issue 4, pp. 2342 - 2359

This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear...

oracle inequalities and semidefinite programming | matrix completion | Noise | Measurement uncertainty | Convex optimization | Minimization | norm of random matrices | Linear matrix inequalities | Noise measurement | Sparse matrices | Compressed sensing | Dantzig selector | COMPUTER SCIENCE, INFORMATION SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Matrices | Research | Information theory | Error correction & detection | Matrix | Information processing

oracle inequalities and semidefinite programming | matrix completion | Noise | Measurement uncertainty | Convex optimization | Minimization | norm of random matrices | Linear matrix inequalities | Noise measurement | Sparse matrices | Compressed sensing | Dantzig selector | COMPUTER SCIENCE, INFORMATION SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Matrices | Research | Information theory | Error correction & detection | Matrix | Information processing

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2010, Volume 20, Issue 4, pp. 1956 - 1982

This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This...

Nuclear norm minimization | Lagrange dual function | Linearized Bregman iteration | Singular value thresholding | Uzawa's algorithm | Matrix completion | SPARSITY | MATHEMATICS, APPLIED | matrix completion | APPROXIMATION | SIGNAL RECOVERY | DECOMPOSITION | RESTORATION | nuclear norm minimization | IMAGE | singular value thresholding | LINEARIZED BREGMAN ITERATIONS | L-MINIMIZATION | linearized Bregman iteration | INVERSE PROBLEMS | CONVERGENCE | Studies | Matrix | Algorithms | Convex analysis | Approximations | Approximation | Mathematical analysis | Norms | Matrices | Matrix methods | Optimization | Convergence

Nuclear norm minimization | Lagrange dual function | Linearized Bregman iteration | Singular value thresholding | Uzawa's algorithm | Matrix completion | SPARSITY | MATHEMATICS, APPLIED | matrix completion | APPROXIMATION | SIGNAL RECOVERY | DECOMPOSITION | RESTORATION | nuclear norm minimization | IMAGE | singular value thresholding | LINEARIZED BREGMAN ITERATIONS | L-MINIMIZATION | linearized Bregman iteration | INVERSE PROBLEMS | CONVERGENCE | Studies | Matrix | Algorithms | Convex analysis | Approximations | Approximation | Mathematical analysis | Norms | Matrices | Matrix methods | Optimization | Convergence

Journal Article