Linear algebra and its applications, ISSN 0024-3795, 2004, Volume 385, Issue 1-3, pp. 305 - 334

We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the...

Positive semidefinite matrix | Matrix inequality | Geometric mean | Matrix square root | Spectral radius | matrix square root | MATHEMATICS, APPLIED | positive semidefinite matrix | spectral radius | geometric mean | matrix inequality

Positive semidefinite matrix | Matrix inequality | Geometric mean | Matrix square root | Spectral radius | matrix square root | MATHEMATICS, APPLIED | positive semidefinite matrix | spectral radius | geometric mean | matrix inequality

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 11/2017, Volume 533, pp. 418 - 427

... (Schur) geometric mean of the sets Ψ1,…,Ψm.

Joint and generalized spectral radius | Bounded sets of operators | Hadamard–Schur geometric mean | Non-negative matrices | Hadamard–Schur product | Positive kernel operators | MATHEMATICS, APPLIED | SEQUENCE-SPACES | INEQUALITIES | THEOREM | NORMS | NONNEGATIVE MATRICES | FORMULA | MATHEMATICS | SEMIGROUPS | PRODUCTS | VERSION | Hadamard-Schur geometric mean | Hadamard-Schur product | Mechanical engineering

Joint and generalized spectral radius | Bounded sets of operators | Hadamard–Schur geometric mean | Non-negative matrices | Hadamard–Schur product | Positive kernel operators | MATHEMATICS, APPLIED | SEQUENCE-SPACES | INEQUALITIES | THEOREM | NORMS | NONNEGATIVE MATRICES | FORMULA | MATHEMATICS | SEMIGROUPS | PRODUCTS | VERSION | Hadamard-Schur geometric mean | Hadamard-Schur product | Mechanical engineering

Journal Article

IEEE journal of selected topics in applied earth observations and remote sensing, ISSN 2151-1535, 2012, Volume 5, Issue 2, pp. 354 - 379

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher...

image processing | hyperspectral remote sensing | Mortar | linear mixture | Educational institutions | remote sensing | unmixing | Vectors | sparsity | image analysis | spectroscopy | imaging spectroscopy | inverse problems | nonlinear mixtures | machine learning algorithms | Hyperspectral imaging | pattern recognition | ALGORITHM | SPATIAL CLASSIFICATION | INDEPENDENT COMPONENT ANALYSIS | BAND SELECTION | IMAGE | DIMENSIONALITY REDUCTION | FOOD QUALITY | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | ENGINEERING, ELECTRICAL & ELECTRONIC | ENDMEMBER EXTRACTION | GEOGRAPHY, PHYSICAL | SPECTRAL MIXTURE ANALYSIS | REFLECTANCE SPECTROSCOPY | Studies | Spectrum analysis | Algorithms | Engineering Sciences | Computer Science | Signal and Image processing

image processing | hyperspectral remote sensing | Mortar | linear mixture | Educational institutions | remote sensing | unmixing | Vectors | sparsity | image analysis | spectroscopy | imaging spectroscopy | inverse problems | nonlinear mixtures | machine learning algorithms | Hyperspectral imaging | pattern recognition | ALGORITHM | SPATIAL CLASSIFICATION | INDEPENDENT COMPONENT ANALYSIS | BAND SELECTION | IMAGE | DIMENSIONALITY REDUCTION | FOOD QUALITY | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | ENGINEERING, ELECTRICAL & ELECTRONIC | ENDMEMBER EXTRACTION | GEOGRAPHY, PHYSICAL | SPECTRAL MIXTURE ANALYSIS | REFLECTANCE SPECTROSCOPY | Studies | Spectrum analysis | Algorithms | Engineering Sciences | Computer Science | Signal and Image processing

Journal Article

Linear and Multilinear Algebra, ISSN 0308-1087, 11/2019, Volume 67, Issue 11, pp. 2159 - 2172

... and the joint spectral radius , where denotes the Hadamard (Schur) geometric mean of the sets Ψ and Σ...

essential spectral radius | positive kernel operators | measure of noncompactness | non-negative matrices | bounded sets of operators | Hadamard-Schur geometric mean | joint and generalized spectral radius | Hadamard-Schur product | Hadamard–Schur geometric mean | Hadamard–Schur product | MATHEMATICS | SEMIGROUPS | PRODUCTS | NORM | Kernels | Operators (mathematics) | Norms | Spectra | Function space | Inequalities

essential spectral radius | positive kernel operators | measure of noncompactness | non-negative matrices | bounded sets of operators | Hadamard-Schur geometric mean | joint and generalized spectral radius | Hadamard-Schur product | Hadamard–Schur geometric mean | Hadamard–Schur product | MATHEMATICS | SEMIGROUPS | PRODUCTS | NORM | Kernels | Operators (mathematics) | Norms | Spectra | Function space | Inequalities

Journal Article

Water Resources Research, ISSN 0043-1397, 06/2016, Volume 52, Issue 6, pp. 4321 - 4337

We estimate parameters from the Katz and Thompson permeability model using laboratory complex electrical conductivity (CC) and nuclear magnetic resonance (NMR)...

induced polarization | complex conductivity | nuclear magnetic resonance | permeability | ROCKS | WATER RESOURCES | SPECTRAL-INDUCED POLARIZATION | POROSIMETRY | RELAXATION-TIME | ENVIRONMENTAL SCIENCES | TRANSPORT | AQUIFER | HYDRAULIC CONDUCTIVITY | NMR RELAXATION | POROUS-MEDIA | LIMNOLOGY | WATER | Conductivity | Nuclear magnetic resonance--NMR | Permeability | Laboratories | Hydraulics | Estimates | Root-mean-square errors | Mean square values | Correlation | Parameter estimation | Electrical conductivity | Size | Relaxation time | Sandstones | Sedimentary rocks | Mathematical models | Nuclear magnetic resonance | Sandstone | Construction | Parameters | Magnetic resonance | Geophysics | Formations | Injection | Pressure | Pharynx | Cores | Errors | Electrical resistivity | Length | Magnetic permeability | Scale (ratio) | Capillary pressure | Models | Resonance | Mercury | MATERIALS SCIENCE

induced polarization | complex conductivity | nuclear magnetic resonance | permeability | ROCKS | WATER RESOURCES | SPECTRAL-INDUCED POLARIZATION | POROSIMETRY | RELAXATION-TIME | ENVIRONMENTAL SCIENCES | TRANSPORT | AQUIFER | HYDRAULIC CONDUCTIVITY | NMR RELAXATION | POROUS-MEDIA | LIMNOLOGY | WATER | Conductivity | Nuclear magnetic resonance--NMR | Permeability | Laboratories | Hydraulics | Estimates | Root-mean-square errors | Mean square values | Correlation | Parameter estimation | Electrical conductivity | Size | Relaxation time | Sandstones | Sedimentary rocks | Mathematical models | Nuclear magnetic resonance | Sandstone | Construction | Parameters | Magnetic resonance | Geophysics | Formations | Injection | Pressure | Pharynx | Cores | Errors | Electrical resistivity | Length | Magnetic permeability | Scale (ratio) | Capillary pressure | Models | Resonance | Mercury | MATERIALS SCIENCE

Journal Article

Biomechanics and modeling in mechanobiology, ISSN 1617-7940, 2017, Volume 17, Issue 2, pp. 351 - 366

.... However, challenges remain in the development of robust means for the quantification and representation of MV leaflet geometry...

Biomedical Engineering | Engineering | Mitral valve | Very fast image in-painting | High-fidelity model | Mitral valve repair | Anatomical accuracy | Biological and Medical Physics, Biophysics | Theoretical and Applied Mechanics | Multi-resolution model | Sparse spectral analysis | ENGINEERING, BIOMEDICAL | STRUCTURAL CONSTITUTIVE MODEL | FINITE-ELEMENT IMPLEMENTATION | REPAIR | ANNULOPLASTY | BIOPHYSICS | SHRINKAGE | REGURGITATION | DISEASE | SEGMENTATION | SIMULATOR | TISSUES | Mitral Valve - anatomy & histology | Biomechanical Phenomena | Animals | Image Processing, Computer-Assisted | Models, Cardiovascular | Male | Sheep | X-Ray Microtomography | Heart | Computer-generated environments | Computer simulation | Analysis | Biomedical engineering | Medical treatment | Spatial discrimination | Fourier analysis | Geometric accuracy | Heart valves | Blood flow | Computer programs | Geometry | Valve leaflets | Simulation | Computation | Computer applications | Mathematical models | Rheumatic heart disease | Two dimensional analysis | Spatial resolution | Heart diseases | Surface geometry | sparse spectral analysis | high-fidelity model | mitral valve | multi-resolution model | mitral valve repair | anatomical accuracy

Biomedical Engineering | Engineering | Mitral valve | Very fast image in-painting | High-fidelity model | Mitral valve repair | Anatomical accuracy | Biological and Medical Physics, Biophysics | Theoretical and Applied Mechanics | Multi-resolution model | Sparse spectral analysis | ENGINEERING, BIOMEDICAL | STRUCTURAL CONSTITUTIVE MODEL | FINITE-ELEMENT IMPLEMENTATION | REPAIR | ANNULOPLASTY | BIOPHYSICS | SHRINKAGE | REGURGITATION | DISEASE | SEGMENTATION | SIMULATOR | TISSUES | Mitral Valve - anatomy & histology | Biomechanical Phenomena | Animals | Image Processing, Computer-Assisted | Models, Cardiovascular | Male | Sheep | X-Ray Microtomography | Heart | Computer-generated environments | Computer simulation | Analysis | Biomedical engineering | Medical treatment | Spatial discrimination | Fourier analysis | Geometric accuracy | Heart valves | Blood flow | Computer programs | Geometry | Valve leaflets | Simulation | Computation | Computer applications | Mathematical models | Rheumatic heart disease | Two dimensional analysis | Spatial resolution | Heart diseases | Surface geometry | sparse spectral analysis | high-fidelity model | mitral valve | multi-resolution model | mitral valve repair | anatomical accuracy

Journal Article

The Annals of statistics, ISSN 0090-5364, 2010, Volume 38, Issue 2, pp. 1034 - 1070

.... We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases...

Ergodic theory | Integers | Statistical variance | Mathematical theorems | Markov chains | Standard error | Interval estimators | Estimators | Consistent estimators | Estimation methods | Markov chain | Batch means | Monte Carlo | Standard errors | Spectral methods | batch means | spectral methods | RANDOM EFFECTS MODEL | DATA AUGMENTATION | EXPLORING POSTERIOR DISTRIBUTIONS | STATISTICS & PROBABILITY | METROPOLIS ALGORITHMS | SIMULATION OUTPUT ANALYSIS | standard errors | STRONG CONSISTENCY | INDEPENDENT RVS | PARTIAL SUMS | GEOMETRIC ERGODICITY | CONVERGENCE-RATES | 62M15 | 60J22

Ergodic theory | Integers | Statistical variance | Mathematical theorems | Markov chains | Standard error | Interval estimators | Estimators | Consistent estimators | Estimation methods | Markov chain | Batch means | Monte Carlo | Standard errors | Spectral methods | batch means | spectral methods | RANDOM EFFECTS MODEL | DATA AUGMENTATION | EXPLORING POSTERIOR DISTRIBUTIONS | STATISTICS & PROBABILITY | METROPOLIS ALGORITHMS | SIMULATION OUTPUT ANALYSIS | standard errors | STRONG CONSISTENCY | INDEPENDENT RVS | PARTIAL SUMS | GEOMETRIC ERGODICITY | CONVERGENCE-RATES | 62M15 | 60J22

Journal Article

Journal of Complex Networks, ISSN 2051-1310, 2018, Volume 6, Issue 2, pp. 157 - 172

.... This formulation provides a geometric interpretation of Markov Stability in terms of a time-dependent spectral embedding, where the Markov time acts as an inhomogeneous geometric resolution factor...

Partitioning algorithms | Markov stability | Multiscale community detection | Modularity | Spectral methods

Partitioning algorithms | Markov stability | Multiscale community detection | Modularity | Spectral methods

Journal Article

The Visual Computer, ISSN 0178-2789, 5/2009, Volume 25, Issue 5, pp. 667 - 675

.... Our method extracts salient geometric feature points in the Laplace–Beltrami spectral domain instead of usual spatial domains...

Computer Graphics | Spectral geometry | Computer Science | Image Processing and Computer Vision | Shape matching | Artificial Intelligence (incl. Robotics) | Geometric analysis | COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Graphics | Spectral geometry | Computer Science | Image Processing and Computer Vision | Shape matching | Artificial Intelligence (incl. Robotics) | Geometric analysis | COMPUTER SCIENCE, SOFTWARE ENGINEERING

Journal Article

ACM Transactions on Graphics (TOG), ISSN 0730-0301, 07/2019, Volume 38, Issue 4, pp. 1 - 13

We introduce a novel approach to measure the behavior of a geometric operator before and after...

geometry processing | numerical coarsening | spectral geometry | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SPARSE

geometry processing | numerical coarsening | spectral geometry | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SPARSE

Journal Article

Linear and Multilinear Algebra, ISSN 0308-1087, 08/2019, Volume 67, Issue 8, pp. 1637 - 1652

... on sequence spaces, or of the Hadamard geometric mean and ordinary products of positive kernel operators on Banach function spaces...

nonnegative matrices | Secondary: 47A10 | positive kernel operators | Hadamard-Schur weighted geometric mean | Banach function spaces | spectral radius | Primary: 47B65 | MATHEMATICS | PRODUCTS | BOUNDED SETS | JOINT | Kernels | Operators (mathematics) | Function space | Inequalities

nonnegative matrices | Secondary: 47A10 | positive kernel operators | Hadamard-Schur weighted geometric mean | Banach function spaces | spectral radius | Primary: 47B65 | MATHEMATICS | PRODUCTS | BOUNDED SETS | JOINT | Kernels | Operators (mathematics) | Function space | Inequalities

Journal Article

Remote Sensing of Environment, ISSN 0034-4257, 09/2012, Volume 124, pp. 384 - 393

... (mean, dominant, and Lorey's height) across Canada's northern forests by integrating lidar data (representing 0.27% of the study area...

Lidar plots | Large-area | Tree height | Li–Strahler geometric-optical model | Landsat | Li-Strahler geometric-optical model | QUICKBIRD IMAGERY | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | HEIGHT | STRUCTURAL-CHANGE DETECTION | ENVIRONMENTAL SCIENCES | FOREST CANOPY | REMOTE SENSING | INVENTORY | AIRBORNE | RESOLUTION SATELLITE IMAGERY | SPECTRAL MIXTURE ANALYSIS | CANOPY REFLECTANCE MODEL | VEGETATION | Earth resources technology satellites | Analysis | Remote sensing

Lidar plots | Large-area | Tree height | Li–Strahler geometric-optical model | Landsat | Li-Strahler geometric-optical model | QUICKBIRD IMAGERY | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | HEIGHT | STRUCTURAL-CHANGE DETECTION | ENVIRONMENTAL SCIENCES | FOREST CANOPY | REMOTE SENSING | INVENTORY | AIRBORNE | RESOLUTION SATELLITE IMAGERY | SPECTRAL MIXTURE ANALYSIS | CANOPY REFLECTANCE MODEL | VEGETATION | Earth resources technology satellites | Analysis | Remote sensing

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 10/2017, Volume 531, pp. 268 - 280

.... We show that the weighted means satisfying the arithmetic-geometric-harmonic mean inequalities are the multivariate Lie-Trotter means...

Arithmetic-geometric-harmonic mean inequalities | Lie-Trotter formula | Inductive mean | Geometric mean | Spectral geometric mean | MATHEMATICS | MATHEMATICS, APPLIED

Arithmetic-geometric-harmonic mean inequalities | Lie-Trotter formula | Inductive mean | Geometric mean | Spectral geometric mean | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 11/2012, Volume 437, Issue 9, pp. 2159 - 2172

In this paper we provide a new class of (metric) geometric means of positive definite matrices varying over Hermitian unitary matrices...

Hermitian unitary matrix | Hadamard space | Geometric mean | Factorization | Spectral geometric mean | Positive definite matrix | MATHEMATICS | MATHEMATICS, APPLIED | SUBMANIFOLDS | ANDO-LI-MATHIAS

Hermitian unitary matrix | Hadamard space | Geometric mean | Factorization | Spectral geometric mean | Positive definite matrix | MATHEMATICS | MATHEMATICS, APPLIED | SUBMANIFOLDS | ANDO-LI-MATHIAS

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 2007, Volume 427, Issue 2, pp. 190 - 196

...–Trotter formula for weighted Log-Euclidean geometric means of several positive definite operators is given in terms of Sagae...

Lie–Trotter formula | Log-Euclidean mean | Sagae–Tanabe mean | Geometric mean | Positive definite operator | Spectral geometric mean | Lie-Trotter formula | Sagae-Tanabe mean | MATHEMATICS | spectral geometric mean | MATHEMATICS, APPLIED | sagae-tanabe mean | INEQUALITIES | positive definite operator | geometric mean | log-euclidean mean

Lie–Trotter formula | Log-Euclidean mean | Sagae–Tanabe mean | Geometric mean | Positive definite operator | Spectral geometric mean | Lie-Trotter formula | Sagae-Tanabe mean | MATHEMATICS | spectral geometric mean | MATHEMATICS, APPLIED | sagae-tanabe mean | INEQUALITIES | positive definite operator | geometric mean | log-euclidean mean

Journal Article

Proceedings of the National Academy of Sciences - PNAS, ISSN 1091-6490, 2005, Volume 102, Issue 21, pp. 7426 - 7431

We provide a framework for structural multiscale geometric organization of graphs and subsets of Rn...

Datasets | Geometry | Embeddings | Physical Sciences | Spectral theory | Eigenfunctions | Harmonic analysis | Laplacians | Mathematical functions | Euclidean space | Spectral graph theory | EIGENMAPS | DIMENSIONALITY REDUCTION | MULTIDISCIPLINARY SCIENCES | Series, Geometric | Analysis

Datasets | Geometry | Embeddings | Physical Sciences | Spectral theory | Eigenfunctions | Harmonic analysis | Laplacians | Mathematical functions | Euclidean space | Spectral graph theory | EIGENMAPS | DIMENSIONALITY REDUCTION | MULTIDISCIPLINARY SCIENCES | Series, Geometric | Analysis

Journal Article

Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, 06/2014, Volume 57, Issue 2, pp. 565 - 571

...–Heinz inequality means that A ≤ B implies that Aa ≦ Ba. We show that A ≤ B if and only if (A + λ)a ≦ (B + λ)a for every λ > 0. We then apply this to the geometric mean and spectral order.

Loewner-Heinz inequality | geometric mean | spectral order | MATHEMATICS | Geometry | Applied mathematics | Algebra | Operators | Spectra | Real numbers | Mathematical analysis | Inequalities

Loewner-Heinz inequality | geometric mean | spectral order | MATHEMATICS | Geometry | Applied mathematics | Algebra | Operators | Spectra | Real numbers | Mathematical analysis | Inequalities

Journal Article