2006, Foundations and trends in communications and information theory, ISBN 1933019239, Volume 2, issue 3 (2006)., Issue v 2 3, 155-239, ix, 93

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner...

Toeplitz matrices | Matrices | Theorems (Mathematics) | Research | Mathematical research | Analysis of covariance

Toeplitz matrices | Matrices | Theorems (Mathematics) | Research | Mathematical research | Analysis of covariance

Book

2015, Cambridge series in statistical and probabilistic mathematics, ISBN 1107065178, xiv, 308

Book

The Annals of statistics, ISSN 0090-5364, 2008, Volume 36, Issue 1, pp. 199 - 227

This paper considers estimating a covariance matrix of p variables from n observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance...

Approximation | Covariance | Spectral energy distribution | Eigenvalues | Population estimates | White noise | Traffic estimation | Covariance matrices | Estimators | Oracles | Cholesky decomposition | Banding | Covariance matrix | Regularization | EIGENVALUE | regularization | banding | STATISTICS & PROBABILITY | LIMIT | SELECTION | covariance matrix | 62G09 | 62F12 | 62H12

Approximation | Covariance | Spectral energy distribution | Eigenvalues | Population estimates | White noise | Traffic estimation | Covariance matrices | Estimators | Oracles | Cholesky decomposition | Banding | Covariance matrix | Regularization | EIGENVALUE | regularization | banding | STATISTICS & PROBABILITY | LIMIT | SELECTION | covariance matrix | 62G09 | 62F12 | 62H12

Journal Article

The Annals of applied probability, ISSN 1050-5164, 6/2014, Volume 24, Issue 3, pp. 935 - 1001

In this paper we prove the universality of covariance matrices of the form HN × N = X†X where X is an M × N rectangular matrix with independent real valued entries xij satisfying Exij...

Brownian motion | Spectral theory | Eigenvalues | Matrices | Random variables | Universality | Covariance matrices | Statistics | Greens function | Dyson Brownian motion | Covariance matrix | Marcenko-Pastur law | Tracy-Widom law | universality | ENSEMBLES | STATISTICS & PROBABILITY | BULK UNIVERSALITY | LIMIT | SPECTRAL STATISTICS | MODELS | LOCAL EIGENVALUE STATISTICS | CONVERGENCE | GENERALIZED WIGNER MATRICES | EDGE | SMALLEST EIGENVALUE | 15B52 | Marcenko–Pastur law | 82B44 | Tracy–Widom law

Brownian motion | Spectral theory | Eigenvalues | Matrices | Random variables | Universality | Covariance matrices | Statistics | Greens function | Dyson Brownian motion | Covariance matrix | Marcenko-Pastur law | Tracy-Widom law | universality | ENSEMBLES | STATISTICS & PROBABILITY | BULK UNIVERSALITY | LIMIT | SPECTRAL STATISTICS | MODELS | LOCAL EIGENVALUE STATISTICS | CONVERGENCE | GENERALIZED WIGNER MATRICES | EDGE | SMALLEST EIGENVALUE | 15B52 | Marcenko–Pastur law | 82B44 | Tracy–Widom law

Journal Article

The Annals of statistics, ISSN 0090-5364, 2011, Volume 39, Issue 2, pp. 887 - 930

... × T -matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the number mT of unknown entries can be much larger than the sample size N...

Integers | Minimax | Sample size | Analytical estimating | Matrices | Entropy | Random variables | Regression analysis | Covariance matrices | Estimators | Sparse recovery | Empirical process | Quasi-convex Schatten class embeddings | Schatten norm | Penalized least-squares estimator | High-dimensional low-rank matrices | CONSISTENCY | penalized least-squares estimator | empirical process | quasi-convex Schatten class embeddings | sparse recovery | STATISTICS & PROBABILITY | TRACE-NORM | SELECTION | AGGREGATION | ENTROPY | Probability | Mathematics | 62G05 | 62F10

Integers | Minimax | Sample size | Analytical estimating | Matrices | Entropy | Random variables | Regression analysis | Covariance matrices | Estimators | Sparse recovery | Empirical process | Quasi-convex Schatten class embeddings | Schatten norm | Penalized least-squares estimator | High-dimensional low-rank matrices | CONSISTENCY | penalized least-squares estimator | empirical process | quasi-convex Schatten class embeddings | sparse recovery | STATISTICS & PROBABILITY | TRACE-NORM | SELECTION | AGGREGATION | ENTROPY | Probability | Mathematics | 62G05 | 62F10

Journal Article

Journal of the American Statistical Association, ISSN 1537-274X, 2010, Volume 105, Issue 490, pp. 810 - 819

We propose tests for sphericity and identity of high-dimensional covariance matrices...

Gene-set testing | Identity test | Sphericity test | High data dimension | Large p, small n | Statistical variance | Null hypothesis | Covariance | Sample size | High dimensional spaces | Theory and Methods | Genes | Eigenvalues | Covariance matrices | Genotypes | Estimators | small n | HYPOTHESIS TESTS | STATISTICS & PROBABILITY | MODEL | HIGHER CRITICISM | FALSE DISCOVERY RATE | GENE-EXPRESSION | NORMALIZATION | Large p | MICROARRAY DATA | Nonparametric tests | Usage | Matrices | Analysis | Genetic screening

Gene-set testing | Identity test | Sphericity test | High data dimension | Large p, small n | Statistical variance | Null hypothesis | Covariance | Sample size | High dimensional spaces | Theory and Methods | Genes | Eigenvalues | Covariance matrices | Genotypes | Estimators | small n | HYPOTHESIS TESTS | STATISTICS & PROBABILITY | MODEL | HIGHER CRITICISM | FALSE DISCOVERY RATE | GENE-EXPRESSION | NORMALIZATION | Large p | MICROARRAY DATA | Nonparametric tests | Usage | Matrices | Analysis | Genetic screening

Journal Article

IEEE Transactions on Signal Processing, ISSN 1053-587X, 03/2009, Volume 57, Issue 3, pp. 878 - 891

We propose a new low-complexity approximate joint diagonalization (AJD) algorithm, which incorporates nontrivial block-diagonal weight matrices into a weighted least-squares (WLS) AJD criterion...

auto regressive processes | Source separation | Symmetric matrices | Predictive models | Biomedical computing | Blind source separation | Covariance matrix | blind source separation (BSS) | Signal processing algorithms | Approximate joint diagonalization (AJD) | Iterative algorithms | Large-scale systems | nonstationary random processes | Autoregressive processes | Nonstationary random processes | Auto regressive processes | Blind source separation (BSS) | MIXTURE | BLIND SOURCE SEPARATION | ALGORITHMS | auto-regressive processes | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Asymptotes | Stochastic processes | Analysis | Autoregression (Statistics) | Signal processing | Research | Studies | Algorithms | Approximation | Asymptotic properties | Mathematical analysis | Blocking | Matrices | Matrix methods | Optimization

auto regressive processes | Source separation | Symmetric matrices | Predictive models | Biomedical computing | Blind source separation | Covariance matrix | blind source separation (BSS) | Signal processing algorithms | Approximate joint diagonalization (AJD) | Iterative algorithms | Large-scale systems | nonstationary random processes | Autoregressive processes | Nonstationary random processes | Auto regressive processes | Blind source separation (BSS) | MIXTURE | BLIND SOURCE SEPARATION | ALGORITHMS | auto-regressive processes | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Asymptotes | Stochastic processes | Analysis | Autoregression (Statistics) | Signal processing | Research | Studies | Algorithms | Approximation | Asymptotic properties | Mathematical analysis | Blocking | Matrices | Matrix methods | Optimization

Journal Article

Journal of the Royal Statistical Society. Series B, Statistical methodology, ISSN 1369-7412, 2013, Volume 75, Issue 4, pp. 603 - 680

.... By assuming a sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors...

Covariance | Threshing | Eigenvalues | Principal components analysis | Poetry | Covariance matrices | Financial portfolios | Estimators | Consistent estimators | Estimation methods | Cross‐sectional correlation | Diverging eigenvalues | Thresholding | Sparse matrix | Approximate factor model | Low rank matrix | High dimensionality | Unknown factors | Principal components | Cross-sectional correlation | LARGEST EIGENVALUE | COMPONENTS-ANALYSIS | STATISTICS & PROBABILITY | HIGH-DIMENSION | PORTFOLIO SELECTION | FALSE DISCOVERY | CONSISTENCY | OPTIMAL RATES | DYNAMIC-FACTOR MODEL | MATRIX DECOMPOSITION | LARGE NUMBER | Studies | Mathematical models | Statistical analysis | Matrix | Statistics | Approximation | Asymptotic properties | Complement | Covariance matrix | Convergence | thresholding | diverging eigenvalues | sparse matrix | principal components | approximate factor model | unknown factors | High-dimensionality | cross-sectional correlation | low-rank matrix

Covariance | Threshing | Eigenvalues | Principal components analysis | Poetry | Covariance matrices | Financial portfolios | Estimators | Consistent estimators | Estimation methods | Cross‐sectional correlation | Diverging eigenvalues | Thresholding | Sparse matrix | Approximate factor model | Low rank matrix | High dimensionality | Unknown factors | Principal components | Cross-sectional correlation | LARGEST EIGENVALUE | COMPONENTS-ANALYSIS | STATISTICS & PROBABILITY | HIGH-DIMENSION | PORTFOLIO SELECTION | FALSE DISCOVERY | CONSISTENCY | OPTIMAL RATES | DYNAMIC-FACTOR MODEL | MATRIX DECOMPOSITION | LARGE NUMBER | Studies | Mathematical models | Statistical analysis | Matrix | Statistics | Approximation | Asymptotic properties | Complement | Covariance matrix | Convergence | thresholding | diverging eigenvalues | sparse matrix | principal components | approximate factor model | unknown factors | High-dimensionality | cross-sectional correlation | low-rank matrix

Journal Article

The Annals of statistics, ISSN 0090-5364, 2012, Volume 40, Issue 2, pp. 1024 - 1060

Many statistical applications require an estimate of a covariance matrix and/or its...

Monte Carlo methods | Sample size | Eigenvalues | Population estimates | Eigenvectors | Covariance matrices | Rotation | Estimators | Consistent estimators | Oracles | Rotation equivariance | Large-dimensional asymptotics | Nonlinear shrinkage | EIGENVALUES | rotation equivariance | nonlinear shrinkage | MODELS | LIMITING SPECTRAL DISTRIBUTION | STATISTICS & PROBABILITY | 15A52 | 62G20 | 62H12

Monte Carlo methods | Sample size | Eigenvalues | Population estimates | Eigenvectors | Covariance matrices | Rotation | Estimators | Consistent estimators | Oracles | Rotation equivariance | Large-dimensional asymptotics | Nonlinear shrinkage | EIGENVALUES | rotation equivariance | nonlinear shrinkage | MODELS | LIMITING SPECTRAL DISTRIBUTION | STATISTICS & PROBABILITY | 15A52 | 62G20 | 62H12

Journal Article

The Annals of statistics, ISSN 0090-5364, 2015, Volume 43, Issue 1, pp. 177 - 214

Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries...

Latent space model | Sochastic blockmodel | Low rank matrices | Covariance matrix | Matrix estimation | Graphons | Distance matrix | Matrix completion | Singular value decomposition | NUMBER | graphons | STOCHASTIC BLOCKMODELS | STATISTICS & PROBABILITY | ALGORITHMS | MODEL | latent space model | distance matrix | sochastic blockmodel | covariance matrix | GRAPHS | PREDICTION | low rank matrices | matrix estimation | PENALIZATION | NORM | COMPLETION | singular value decomposition | 62G05 | 05C99 | 60B20 | 62F12

Latent space model | Sochastic blockmodel | Low rank matrices | Covariance matrix | Matrix estimation | Graphons | Distance matrix | Matrix completion | Singular value decomposition | NUMBER | graphons | STOCHASTIC BLOCKMODELS | STATISTICS & PROBABILITY | ALGORITHMS | MODEL | latent space model | distance matrix | sochastic blockmodel | covariance matrix | GRAPHS | PREDICTION | low rank matrices | matrix estimation | PENALIZATION | NORM | COMPLETION | singular value decomposition | 62G05 | 05C99 | 60B20 | 62F12

Journal Article

The Annals of statistics, ISSN 0090-5364, 6/2011, Volume 39, Issue 3, pp. 1496 - 1525

Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing...

Handedness | Covariance | Logical givens | Law of large numbers | Eigenvalues | Matrices | Random variables | Sampling distributions | Covariance matrices | Statistics | Random matrix | Compressed sensing matrix | Coherence | Sample correlation matrix | Covariance structure | Limiting distribution | Moderate deviations | Stein method | Maxima | Mutual incoherence property | Chen | random matrix | LARGEST ENTRIES | covariance structure | mutual incoherence property | DANTZIG SELECTOR | coherence | LARGEST EIGENVALUE | DIMENSIONAL FEATURE SPACE | Chen-Stein method | STABLE RECOVERY | STATISTICS & PROBABILITY | SPARSE SIGNALS | moderate deviations | law of large numbers | maxima | ASYMPTOTIC-DISTRIBUTION | sample correlation matrix | SAMPLE CORRELATION-MATRICES | MULTIVARIATE-ANALYSIS | limiting distribution | compressed sensing matrix | STATISTICAL ESTIMATION | Chen–Stein method | 62H10 | 62H12 | 60F05 | 60F15

Handedness | Covariance | Logical givens | Law of large numbers | Eigenvalues | Matrices | Random variables | Sampling distributions | Covariance matrices | Statistics | Random matrix | Compressed sensing matrix | Coherence | Sample correlation matrix | Covariance structure | Limiting distribution | Moderate deviations | Stein method | Maxima | Mutual incoherence property | Chen | random matrix | LARGEST ENTRIES | covariance structure | mutual incoherence property | DANTZIG SELECTOR | coherence | LARGEST EIGENVALUE | DIMENSIONAL FEATURE SPACE | Chen-Stein method | STABLE RECOVERY | STATISTICS & PROBABILITY | SPARSE SIGNALS | moderate deviations | law of large numbers | maxima | ASYMPTOTIC-DISTRIBUTION | sample correlation matrix | SAMPLE CORRELATION-MATRICES | MULTIVARIATE-ANALYSIS | limiting distribution | compressed sensing matrix | STATISTICAL ESTIMATION | Chen–Stein method | 62H10 | 62H12 | 60F05 | 60F15

Journal Article

Journal of theoretical probability, ISSN 1572-9230, 2011, Volume 25, Issue 3, pp. 655 - 686

Given a probability distribution in ℝ n with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points...

60H12 | Estimation of covariance matrices | Random matrices with independent columns | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Sample covariance matrices | 46B09 | STATISTICS & PROBABILITY | CONVEX-BODIES

60H12 | Estimation of covariance matrices | Random matrices with independent columns | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Sample covariance matrices | 46B09 | STATISTICS & PROBABILITY | CONVEX-BODIES

Journal Article

Probability Theory and Related Fields, ISSN 0178-8051, 10/2017, Volume 169, Issue 1, pp. 257 - 352

We develop a new method for deriving local laws for a large class of random matrices...

15B52 | Mathematical and Computational Biology | Theoretical, Mathematical and Computational Physics | Operations Research/Decision Theory | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Quantitative Finance | EIGENVALUES | UNIVERSALITY | STATISTICS | FLUCTUATIONS | LIMITING SPECTRAL DISTRIBUTION | TRACY-WIDOM LIMIT | DIMENSIONAL RANDOM MATRICES | SEMICIRCLE LAW | STATISTICS & PROBABILITY | GENERALIZED WIGNER MATRICES | SAMPLE COVARIANCE MATRICES | Anisotropy | Laws, regulations and rules | Data analysis | Deformation | Covariance | Outliers (statistics) | Eigenvalues | Eigenvectors | Phase transitions

15B52 | Mathematical and Computational Biology | Theoretical, Mathematical and Computational Physics | Operations Research/Decision Theory | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Quantitative Finance | EIGENVALUES | UNIVERSALITY | STATISTICS | FLUCTUATIONS | LIMITING SPECTRAL DISTRIBUTION | TRACY-WIDOM LIMIT | DIMENSIONAL RANDOM MATRICES | SEMICIRCLE LAW | STATISTICS & PROBABILITY | GENERALIZED WIGNER MATRICES | SAMPLE COVARIANCE MATRICES | Anisotropy | Laws, regulations and rules | Data analysis | Deformation | Covariance | Outliers (statistics) | Eigenvalues | Eigenvectors | Phase transitions

Journal Article

The Annals of statistics, ISSN 0090-5364, 12/2008, Volume 36, Issue 6, pp. 2717 - 2756

Estimating covariance matrices is a problem of fundamental importance in multivariate statistics...

Covariance | Spectral theory | Threshing | Eigenvalues | Population estimates | Adjacency matrix | Matrices | Covariance matrices | Estimators | Consistent estimators | Eigenvalues of covariance matrices | Random matrix theory | Multivariate statistical analysis | Sparsity | Adjacency matrices | β-sparsity | High-dimensional inference | Correlation matrices | adjacency matrices | STATISTICS & PROBABILITY | sparsity | LIMIT | CLT | EIGENVALUES | high-dimensional inference | correlation matrices | eigenvalues of covariance matrices | random matrix theory | beta-sparsity | multivariate statistical analysis | 62H12

Covariance | Spectral theory | Threshing | Eigenvalues | Population estimates | Adjacency matrix | Matrices | Covariance matrices | Estimators | Consistent estimators | Eigenvalues of covariance matrices | Random matrix theory | Multivariate statistical analysis | Sparsity | Adjacency matrices | β-sparsity | High-dimensional inference | Correlation matrices | adjacency matrices | STATISTICS & PROBABILITY | sparsity | LIMIT | CLT | EIGENVALUES | high-dimensional inference | correlation matrices | eigenvalues of covariance matrices | random matrix theory | beta-sparsity | multivariate statistical analysis | 62H12

Journal Article

The Annals of statistics, ISSN 0090-5364, 2012, Volume 40, Issue 2, pp. 908 - 940

We propose two tests for the equality of covariance matrices between two high-dimensional populations...

Dimensionality | Covariance | Sample size | Genes | Eigenvalues | Unbiased estimators | Covariance matrices | Statistics | Estimators | Martingales | Testing for gene-sets | Large p small n | High-dimensional covariance | Likelihood ratio test | SPARSITY | large p small n | MICROARRAY | LARGEST EIGENVALUE | HYPOTHESIS TESTS | likelihood ratio test | STATISTICS & PROBABILITY | MODEL | CATEGORIES | GENE-EXPRESSION | NORMALIZATION | testing for gene-sets | REGULARIZATION | SELECTION | 62H15 | 62G20 | 62G10

Dimensionality | Covariance | Sample size | Genes | Eigenvalues | Unbiased estimators | Covariance matrices | Statistics | Estimators | Martingales | Testing for gene-sets | Large p small n | High-dimensional covariance | Likelihood ratio test | SPARSITY | large p small n | MICROARRAY | LARGEST EIGENVALUE | HYPOTHESIS TESTS | likelihood ratio test | STATISTICS & PROBABILITY | MODEL | CATEGORIES | GENE-EXPRESSION | NORMALIZATION | testing for gene-sets | REGULARIZATION | SELECTION | 62H15 | 62G20 | 62G10

Journal Article

IEEE transaction on neural networks and learning systems, ISSN 2162-2388, 2017, Volume 28, Issue 12, pp. 2859 - 2871

Data encoded as symmetric positive definite (SPD) matrices frequently arise in many areas of computer vision and machine learning...

Geometry | Learning systems | Symmetric matrices | Dictionaries | Affine-invariant Riemannian (Riem) metric | region covariance descriptors | sparse coding (SC) | Encoding | Sparse matrices | dictionary learning (DL) | Covariance matrices | TENSOR | TEXTURE | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | REGION COVARIANCE | IMAGE | COMPUTER SCIENCE, THEORY & METHODS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Formulations | Computer vision | Euclidean geometry | Matrix methods | Optimization | Learning | Learning algorithms | Mathematical analysis | Coding | Vision | Machine learning | Set theory | Euclidean space

Geometry | Learning systems | Symmetric matrices | Dictionaries | Affine-invariant Riemannian (Riem) metric | region covariance descriptors | sparse coding (SC) | Encoding | Sparse matrices | dictionary learning (DL) | Covariance matrices | TENSOR | TEXTURE | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | REGION COVARIANCE | IMAGE | COMPUTER SCIENCE, THEORY & METHODS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Formulations | Computer vision | Euclidean geometry | Matrix methods | Optimization | Learning | Learning algorithms | Mathematical analysis | Coding | Vision | Machine learning | Set theory | Euclidean space

Journal Article

17.
Full Text
Spectrum Estimation for Large Dimensional Covariance Matrices Using Random Matrix Theory

The Annals of statistics, ISSN 0090-5364, 12/2008, Volume 36, Issue 6, pp. 2757 - 2790

Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics...

Point masses | Spectral theory | Covariance | Eigenvalues |

Point masses | Spectral theory | Covariance | Eigenvalues |