This document discusses a new implementation of k-MLE for mixture modelling of Wishart distributions. It begins with an overview of the Wishart distribution and its properties as an exponential family. It then describes the original k-MLE algorithm and how it can be adapted for Wishart distributions by using Hartigan and Wang's strategy instead of Lloyd's strategy to avoid empty clusters. The document also discusses approaches for initializing the clusters, such as k-means++, and proposes a heuristic to determine the number of clusters on-the-fly rather than fixing k.
After we applied the stochastic Galerkin method to solve stochastic PDE, and solve large linear system, we obtain stochastic solution (random field), which is represented in Karhunen Loeve and PCE basis. No sampling error is involved, only algebraic truncation error. Now we would like to escape classical MCMC path to compute the posterior. We develop an Bayesian* update formula for KLE-PCE coefficients.
2012 mdsp pr12 k means mixture of gaussiannozomuhamada
The document provides the course calendar and lecture plan for a machine learning course. The course calendar lists the class dates and topics to be covered from September to January, including Bayes estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, and clustering algorithms. The lecture plan focuses on clustering methods, including k-means clustering, mixtures of Gaussians models, and using the expectation-maximization (EM) algorithm to estimate the parameters of Gaussian mixture models.
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
Quickselect Under Yaroslavskiy's Dual Pivoting AlgorithmSebastian Wild
I gave this talk at the 24th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2013) on Menorca (Spain).
A paper covering the analyses of this talk (and some more!) has been submitted.
Also, in the talk, I refer to the previous speaker at the conference, my advisor Markus Nebel - corresponding results can be found in an earlier talk of mine:
slideshare.net/sebawild/average-case-analysis-of-java-7s-dual-pivot-quicksort
Check my website for preprints of papers and my other talks:
wwwagak.cs.uni-kl.de/sebastian-wild.html
I am Anthony F. I am a Digital Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, University of Cambridge, UK. I have been helping students with their homework for the past 8 years. I solve assignments related to Digital Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Digital Signal Processing Assignments.
In this talk we consider the question of how to use QMC with an empirical dataset, such as a set of points generated by MCMC. Using ideas from partitioning for parallel computing, we apply recursive bisection to reorder the points, and then interleave the bits of the QMC coordinates to select the appropriate point from the dataset. Numerical tests show that in the case of known distributions this is almost as effective as applying QMC directly to the original distribution. The same recursive bisection can also be used to thin the dataset, by recursively bisecting down to many small subsets of points, and then randomly selecting one point from each subset. This makes it possible to reduce the size of the dataset greatly without significantly increasing the overall error. Co-author: Fei Xie
Markov chain Monte Carlo (MCMC) methods are popularly used in Bayesian computation. However, they need large number of samples for convergence which can become costly when the posterior distribution is expensive to evaluate. Deterministic sampling techniques such as Quasi-Monte Carlo (QMC) can be a useful alternative to MCMC, but the existing QMC methods are mainly developed only for sampling from unit hypercubes. Unfortunately, the posterior distributions can be highly correlated and nonlinear making them occupy very little space inside a hypercube. Thus, most of the samples from QMC can get wasted. The QMC samples can be saved if they can be pulled towards the high probability regions of the posterior distribution using inverse probability transforms. But this can be done only when the distribution function is known, which is rarely the case in Bayesian problems. In this talk, I will discuss a deterministic sampling technique, known as minimum energy designs, which can directly sample from the posterior distributions.
The document discusses methods for performing spatial statistics on large datasets. Standard maximum likelihood estimation is computationally infeasible for datasets with tens of thousands of observations due to the need to compute and store large covariance matrices. The document outlines several approximation methods that can accommodate large datasets, including variogram fitting, pairwise likelihood approximations, independent block approximations, tapering of the covariance function, low-rank approximations using basis functions, and approximations based on stochastic partial differential equations. These methods allow inference for large spatial datasets by avoiding direct computation and storage of large covariance matrices.
After we applied the stochastic Galerkin method to solve stochastic PDE, and solve large linear system, we obtain stochastic solution (random field), which is represented in Karhunen Loeve and PCE basis. No sampling error is involved, only algebraic truncation error. Now we would like to escape classical MCMC path to compute the posterior. We develop an Bayesian* update formula for KLE-PCE coefficients.
2012 mdsp pr12 k means mixture of gaussiannozomuhamada
The document provides the course calendar and lecture plan for a machine learning course. The course calendar lists the class dates and topics to be covered from September to January, including Bayes estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, and clustering algorithms. The lecture plan focuses on clustering methods, including k-means clustering, mixtures of Gaussians models, and using the expectation-maximization (EM) algorithm to estimate the parameters of Gaussian mixture models.
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
Quickselect Under Yaroslavskiy's Dual Pivoting AlgorithmSebastian Wild
I gave this talk at the 24th International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2013) on Menorca (Spain).
A paper covering the analyses of this talk (and some more!) has been submitted.
Also, in the talk, I refer to the previous speaker at the conference, my advisor Markus Nebel - corresponding results can be found in an earlier talk of mine:
slideshare.net/sebawild/average-case-analysis-of-java-7s-dual-pivot-quicksort
Check my website for preprints of papers and my other talks:
wwwagak.cs.uni-kl.de/sebastian-wild.html
I am Anthony F. I am a Digital Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, University of Cambridge, UK. I have been helping students with their homework for the past 8 years. I solve assignments related to Digital Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Digital Signal Processing Assignments.
In this talk we consider the question of how to use QMC with an empirical dataset, such as a set of points generated by MCMC. Using ideas from partitioning for parallel computing, we apply recursive bisection to reorder the points, and then interleave the bits of the QMC coordinates to select the appropriate point from the dataset. Numerical tests show that in the case of known distributions this is almost as effective as applying QMC directly to the original distribution. The same recursive bisection can also be used to thin the dataset, by recursively bisecting down to many small subsets of points, and then randomly selecting one point from each subset. This makes it possible to reduce the size of the dataset greatly without significantly increasing the overall error. Co-author: Fei Xie
Markov chain Monte Carlo (MCMC) methods are popularly used in Bayesian computation. However, they need large number of samples for convergence which can become costly when the posterior distribution is expensive to evaluate. Deterministic sampling techniques such as Quasi-Monte Carlo (QMC) can be a useful alternative to MCMC, but the existing QMC methods are mainly developed only for sampling from unit hypercubes. Unfortunately, the posterior distributions can be highly correlated and nonlinear making them occupy very little space inside a hypercube. Thus, most of the samples from QMC can get wasted. The QMC samples can be saved if they can be pulled towards the high probability regions of the posterior distribution using inverse probability transforms. But this can be done only when the distribution function is known, which is rarely the case in Bayesian problems. In this talk, I will discuss a deterministic sampling technique, known as minimum energy designs, which can directly sample from the posterior distributions.
The document discusses methods for performing spatial statistics on large datasets. Standard maximum likelihood estimation is computationally infeasible for datasets with tens of thousands of observations due to the need to compute and store large covariance matrices. The document outlines several approximation methods that can accommodate large datasets, including variogram fitting, pairwise likelihood approximations, independent block approximations, tapering of the covariance function, low-rank approximations using basis functions, and approximations based on stochastic partial differential equations. These methods allow inference for large spatial datasets by avoiding direct computation and storage of large covariance matrices.
The document provides information about numerical methods topics including:
1) Lagrange's interpolation formula for finding a polynomial that passes through given data points, either equally or unequally spaced. The formula uses divided differences to find the coefficients.
2) Newton's divided difference interpolation formula for unequal intervals that also uses divided differences.
3) The nature of divided differences - for a polynomial of degree n, the nth divided difference is constant.
4) Examples of evaluating divided differences and constructing divided difference tables are given.
I am Irene M. I am a Diffusion Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from California, USA.
I have been helping students with their homework for the past 8 years. I solve assignments related to Diffusion. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Diffusion Assignments.
The document discusses different methods for fitting equations to data, including interpolating polynomials, least squares fitting, and examples of non-polynomial forms. It provides sample MATLAB code for finding an interpolating polynomial that passes through every data point in a sample data set, as well as code for performing a cubic least squares fit. The document concludes by giving practice problems involving fitting equations to saturation property data for water.
This document provides an overview of numerical linear algebra concepts including matrix notation, operations, and solving systems of linear equations using Gaussian elimination. It describes the Gaussian elimination process which involves eliminating variables one by one to obtain an upper triangular system that can then be solved using back substitution. The document notes some pitfalls of naive Gaussian elimination such as division by zero, round-off errors, ill-conditioned systems, and singular systems. It introduces pivoting as a technique to avoid division by zero during the elimination process and calculates the determinant as a byproduct of Gaussian elimination.
The document summarizes sampling methods from Chapter 11 of Bishop's PRML book. It introduces basic sampling algorithms like rejection sampling, importance sampling, and SIR. It then discusses Markov chain Monte Carlo (MCMC) methods which allow sampling from complex distributions using a Markov chain. Specific MCMC methods covered include the Metropolis algorithm, Gibbs sampling, and estimating the partition function using the IP algorithm.
This document provides an outline and overview of key concepts for estimating curves and surfaces from data using basis functions and penalized least squares regression. It discusses representing a curve or surface using basis functions, fitting the coefficients using ordinary least squares, and adding a penalty term to the least squares objective function to produce a smoothed estimate. The smoothing parameter λ controls the tradeoff between fit to the data and smoothness of the estimate. Cross-validation can be used to choose λ.
This document summarizes an exercise involving calculating the inverse of square matrices in three ways: analytically, using LU decomposition, and singular value decomposition. It finds that analytical calculation becomes impractically slow for matrices larger than order 13, while LU decomposition and SVD using GNU Scientific Library functions can calculate the inverse of matrices up to order 350 in under 20 seconds. SVD is found to be slightly more efficient than LU decomposition for higher order matrices. Accuracy is also compared when the input matrix is close to singular, finding SVD returns the most accurate inverse.
The document summarizes an exercise on using random number generators and Monte Carlo techniques. It compares two methods - analytical and accept/reject - for generating a sinusoidal distribution of random numbers from a uniform distribution. Both methods produced very good fits to a sinusoidal distribution when a large number of random numbers were tested, with R^2 values close to 1. The accept/reject method may be more generally applicable since problems can arise with evaluating and inverting aspects of the analytical method. The exercise also used Monte Carlo techniques to model detector resolution in a nuclear physics experiment. Accounting for detector resolution made the calculated distribution less precise and less representative of the true distribution, showing the need for improved detector resolution.
I am Bella A. I am a Statistical Method In Economics Assignment Expert at economicshomeworkhelper.com/. I hold a Ph.D. in Economics. I have been helping students with their homework for the past 9 years. I solve assignments related to Economics Assignment.
Visit economicshomeworkhelper.com/ or email info@economicshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Method In Economics Assignments.
This document provides an overview of linear models for classification. It discusses discriminant functions including linear discriminant analysis and the perceptron algorithm. It also covers probabilistic generative models that model class-conditional densities and priors to estimate posterior probabilities. Probabilistic discriminative models like logistic regression directly model posterior probabilities using maximum likelihood. Iterative reweighted least squares is used to optimize logistic regression since there is no closed-form solution.
This document summarizes an exercise on using computational methods to solve partial differential equations (PDEs). It first discusses using relaxation techniques to solve the Laplace equation for modeling electric fields around parallel plate capacitors. The Jacobi method was used to iteratively calculate potentials on a grid until convergence within a specified tolerance. Smaller tolerances and larger grids required more iterations to converge. Longer capacitor plates produced more uniform fields resembling the theoretical infinite case. The document demonstrates how computational methods can effectively solve important physical problems modeled by PDEs.
This document provides an overview of least-squares regression techniques including:
- Simple linear regression to fit a line to data
- Polynomial regression to fit higher order curves
- Multiple regression to fit surfaces using two or more variables
It discusses calculating regression coefficients, quantifying errors, and performing statistical analysis of the regression results including determining confidence intervals. Examples are provided to demonstrate applying these techniques.
Using several mathematical examples from three different authors in texts from different courses this paper illustrates the easier way to avoid confusions and always get the correct results with the least effort was to use the proposed Excel Gamma function explained in detail for the proper use of the Q(z) and ercf(x) functions in most communication courses. The paper serves as a tutorial and introduction for such functions
This document summarizes a semi-supervised regression method that combines graph Laplacian regularization with cluster ensemble methodology. It proposes using a weighted averaged co-association matrix from the cluster ensemble as the similarity matrix in graph Laplacian regularization. The method (SSR-LRCM) finds a low-rank approximation of the co-association matrix to efficiently solve the regression problem. Experimental results on synthetic and real-world datasets show SSR-LRCM achieves significantly better prediction accuracy than an alternative method, while also having lower computational costs for large datasets. Future work will explore using a hierarchical matrix approximation instead of low-rank.
The document discusses uncertainty quantification (UQ) using quasi-Monte Carlo (QMC) integration methods. It introduces parametric operator equations for modeling input uncertainty in partial differential equations. Both forward and inverse UQ problems are considered. QMC methods like interlaced polynomial lattice rules are discussed for approximating high-dimensional integrals arising in UQ, with convergence rates superior to standard Monte Carlo. Algorithms for single-level and multilevel QMC are presented for solving forward and inverse UQ problems.
The document contains exercises, hints, and solutions for analyzing algorithms from a textbook. It includes problems related to brute force algorithms, sorting algorithms like selection sort and bubble sort, and evaluating polynomials. The solutions analyze the time complexity of different algorithms, such as proving that a brute force polynomial evaluation algorithm is O(n^2) while a modified version is linear time. It also discusses whether sorting algorithms like selection sort and bubble sort preserve the original order of equal elements (i.e. whether they are stable).
This document provides an introduction to Bayesian analysis and probabilistic modeling. It begins with an overview of Bayes' theorem and common probability distributions used in Bayesian modeling like the Bernoulli, binomial, beta, Dirichlet, and multinomial distributions. It then discusses how these distributions can be used in Bayesian modeling for problems like estimating probabilities based on observed data. Specifically, it explains how conjugate prior distributions allow the posterior distribution to be of the same family as the prior. The document concludes by discussing how neural networks can quantify classification uncertainty by outputting evidence for different classes modeled with a Dirichlet distribution.
I am Travis W. I am a Computer Science Assignment Expert at programminghomeworkhelp.com. I hold a Master's in Computer Science, Leeds University. I have been helping students with their homework for the past 9 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
I am Stacy L. I am a Matlab Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, University of Houston. I have been helping students with their homework for the past 9 years. I solve assignments related to Data Analysis.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Data Analysis Assignments.
The document discusses LU decomposition and its applications in numerical linear algebra. It explains that LU decomposition decomposes a matrix A into lower and upper triangular matrices (L and U) such that A = LU. This decomposition allows the efficient solution of linear systems even when the right hand side vector changes. The document also discusses other related topics like matrix inverse, condition number, special matrices, and iterative refinement. It provides examples to illustrate LU decomposition and its use in calculating the inverse of a matrix.
On approximating the Riemannian 1-centerFrank Nielsen
This document discusses algorithms for finding the smallest enclosing ball that fully covers a set of points on a Riemannian manifold. It begins by reviewing Euclidean smallest enclosing ball algorithms, then extends the concept to Riemannian manifolds. Coreset approximations are discussed as well as gradient descent algorithms. The document provides background on Riemannian geometry concepts needed like geodesics, exponential maps, and curvature. Overall, it presents algorithms to generalize the smallest enclosing ball problem to points on Riemannian manifolds.
(slides 8) Visual Computing: Geometry, Graphics, and VisionFrank Nielsen
Those are the slides for the book:
Visual Computing: Geometry, Graphics, and Vision.
by Frank Nielsen (2005)
http://www.sonycsl.co.jp/person/nielsen/visualcomputing/
http://www.amazon.com/Visual-Computing-Geometry-Graphics-Charles/dp/1584504277
The document provides information about numerical methods topics including:
1) Lagrange's interpolation formula for finding a polynomial that passes through given data points, either equally or unequally spaced. The formula uses divided differences to find the coefficients.
2) Newton's divided difference interpolation formula for unequal intervals that also uses divided differences.
3) The nature of divided differences - for a polynomial of degree n, the nth divided difference is constant.
4) Examples of evaluating divided differences and constructing divided difference tables are given.
I am Irene M. I am a Diffusion Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from California, USA.
I have been helping students with their homework for the past 8 years. I solve assignments related to Diffusion. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Diffusion Assignments.
The document discusses different methods for fitting equations to data, including interpolating polynomials, least squares fitting, and examples of non-polynomial forms. It provides sample MATLAB code for finding an interpolating polynomial that passes through every data point in a sample data set, as well as code for performing a cubic least squares fit. The document concludes by giving practice problems involving fitting equations to saturation property data for water.
This document provides an overview of numerical linear algebra concepts including matrix notation, operations, and solving systems of linear equations using Gaussian elimination. It describes the Gaussian elimination process which involves eliminating variables one by one to obtain an upper triangular system that can then be solved using back substitution. The document notes some pitfalls of naive Gaussian elimination such as division by zero, round-off errors, ill-conditioned systems, and singular systems. It introduces pivoting as a technique to avoid division by zero during the elimination process and calculates the determinant as a byproduct of Gaussian elimination.
The document summarizes sampling methods from Chapter 11 of Bishop's PRML book. It introduces basic sampling algorithms like rejection sampling, importance sampling, and SIR. It then discusses Markov chain Monte Carlo (MCMC) methods which allow sampling from complex distributions using a Markov chain. Specific MCMC methods covered include the Metropolis algorithm, Gibbs sampling, and estimating the partition function using the IP algorithm.
This document provides an outline and overview of key concepts for estimating curves and surfaces from data using basis functions and penalized least squares regression. It discusses representing a curve or surface using basis functions, fitting the coefficients using ordinary least squares, and adding a penalty term to the least squares objective function to produce a smoothed estimate. The smoothing parameter λ controls the tradeoff between fit to the data and smoothness of the estimate. Cross-validation can be used to choose λ.
This document summarizes an exercise involving calculating the inverse of square matrices in three ways: analytically, using LU decomposition, and singular value decomposition. It finds that analytical calculation becomes impractically slow for matrices larger than order 13, while LU decomposition and SVD using GNU Scientific Library functions can calculate the inverse of matrices up to order 350 in under 20 seconds. SVD is found to be slightly more efficient than LU decomposition for higher order matrices. Accuracy is also compared when the input matrix is close to singular, finding SVD returns the most accurate inverse.
The document summarizes an exercise on using random number generators and Monte Carlo techniques. It compares two methods - analytical and accept/reject - for generating a sinusoidal distribution of random numbers from a uniform distribution. Both methods produced very good fits to a sinusoidal distribution when a large number of random numbers were tested, with R^2 values close to 1. The accept/reject method may be more generally applicable since problems can arise with evaluating and inverting aspects of the analytical method. The exercise also used Monte Carlo techniques to model detector resolution in a nuclear physics experiment. Accounting for detector resolution made the calculated distribution less precise and less representative of the true distribution, showing the need for improved detector resolution.
I am Bella A. I am a Statistical Method In Economics Assignment Expert at economicshomeworkhelper.com/. I hold a Ph.D. in Economics. I have been helping students with their homework for the past 9 years. I solve assignments related to Economics Assignment.
Visit economicshomeworkhelper.com/ or email info@economicshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Method In Economics Assignments.
This document provides an overview of linear models for classification. It discusses discriminant functions including linear discriminant analysis and the perceptron algorithm. It also covers probabilistic generative models that model class-conditional densities and priors to estimate posterior probabilities. Probabilistic discriminative models like logistic regression directly model posterior probabilities using maximum likelihood. Iterative reweighted least squares is used to optimize logistic regression since there is no closed-form solution.
This document summarizes an exercise on using computational methods to solve partial differential equations (PDEs). It first discusses using relaxation techniques to solve the Laplace equation for modeling electric fields around parallel plate capacitors. The Jacobi method was used to iteratively calculate potentials on a grid until convergence within a specified tolerance. Smaller tolerances and larger grids required more iterations to converge. Longer capacitor plates produced more uniform fields resembling the theoretical infinite case. The document demonstrates how computational methods can effectively solve important physical problems modeled by PDEs.
This document provides an overview of least-squares regression techniques including:
- Simple linear regression to fit a line to data
- Polynomial regression to fit higher order curves
- Multiple regression to fit surfaces using two or more variables
It discusses calculating regression coefficients, quantifying errors, and performing statistical analysis of the regression results including determining confidence intervals. Examples are provided to demonstrate applying these techniques.
Using several mathematical examples from three different authors in texts from different courses this paper illustrates the easier way to avoid confusions and always get the correct results with the least effort was to use the proposed Excel Gamma function explained in detail for the proper use of the Q(z) and ercf(x) functions in most communication courses. The paper serves as a tutorial and introduction for such functions
This document summarizes a semi-supervised regression method that combines graph Laplacian regularization with cluster ensemble methodology. It proposes using a weighted averaged co-association matrix from the cluster ensemble as the similarity matrix in graph Laplacian regularization. The method (SSR-LRCM) finds a low-rank approximation of the co-association matrix to efficiently solve the regression problem. Experimental results on synthetic and real-world datasets show SSR-LRCM achieves significantly better prediction accuracy than an alternative method, while also having lower computational costs for large datasets. Future work will explore using a hierarchical matrix approximation instead of low-rank.
The document discusses uncertainty quantification (UQ) using quasi-Monte Carlo (QMC) integration methods. It introduces parametric operator equations for modeling input uncertainty in partial differential equations. Both forward and inverse UQ problems are considered. QMC methods like interlaced polynomial lattice rules are discussed for approximating high-dimensional integrals arising in UQ, with convergence rates superior to standard Monte Carlo. Algorithms for single-level and multilevel QMC are presented for solving forward and inverse UQ problems.
The document contains exercises, hints, and solutions for analyzing algorithms from a textbook. It includes problems related to brute force algorithms, sorting algorithms like selection sort and bubble sort, and evaluating polynomials. The solutions analyze the time complexity of different algorithms, such as proving that a brute force polynomial evaluation algorithm is O(n^2) while a modified version is linear time. It also discusses whether sorting algorithms like selection sort and bubble sort preserve the original order of equal elements (i.e. whether they are stable).
This document provides an introduction to Bayesian analysis and probabilistic modeling. It begins with an overview of Bayes' theorem and common probability distributions used in Bayesian modeling like the Bernoulli, binomial, beta, Dirichlet, and multinomial distributions. It then discusses how these distributions can be used in Bayesian modeling for problems like estimating probabilities based on observed data. Specifically, it explains how conjugate prior distributions allow the posterior distribution to be of the same family as the prior. The document concludes by discussing how neural networks can quantify classification uncertainty by outputting evidence for different classes modeled with a Dirichlet distribution.
I am Travis W. I am a Computer Science Assignment Expert at programminghomeworkhelp.com. I hold a Master's in Computer Science, Leeds University. I have been helping students with their homework for the past 9 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
I am Stacy L. I am a Matlab Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, University of Houston. I have been helping students with their homework for the past 9 years. I solve assignments related to Data Analysis.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Data Analysis Assignments.
The document discusses LU decomposition and its applications in numerical linear algebra. It explains that LU decomposition decomposes a matrix A into lower and upper triangular matrices (L and U) such that A = LU. This decomposition allows the efficient solution of linear systems even when the right hand side vector changes. The document also discusses other related topics like matrix inverse, condition number, special matrices, and iterative refinement. It provides examples to illustrate LU decomposition and its use in calculating the inverse of a matrix.
On approximating the Riemannian 1-centerFrank Nielsen
This document discusses algorithms for finding the smallest enclosing ball that fully covers a set of points on a Riemannian manifold. It begins by reviewing Euclidean smallest enclosing ball algorithms, then extends the concept to Riemannian manifolds. Coreset approximations are discussed as well as gradient descent algorithms. The document provides background on Riemannian geometry concepts needed like geodesics, exponential maps, and curvature. Overall, it presents algorithms to generalize the smallest enclosing ball problem to points on Riemannian manifolds.
(slides 8) Visual Computing: Geometry, Graphics, and VisionFrank Nielsen
Those are the slides for the book:
Visual Computing: Geometry, Graphics, and Vision.
by Frank Nielsen (2005)
http://www.sonycsl.co.jp/person/nielsen/visualcomputing/
http://www.amazon.com/Visual-Computing-Geometry-Graphics-Charles/dp/1584504277
Voronoi diagrams in information geometry: Statistical Voronoi diagrams and ...Frank Nielsen
1. The document discusses Voronoi diagrams and their applications in statistics and information geometry. It outlines topics including Euclidean Voronoi diagrams, Mahalanobis Voronoi diagrams for discriminant analysis, and Fisher-Hotelling-Rao Voronoi diagrams for curved Riemannian geometries.
2. It also covers estimation techniques like maximum likelihood estimation and exponential families, as well as the Cramér-Rao lower bound for estimator variance. Rao's distance is introduced for measuring distances between populations in the statistical parameter space equipped with the Fisher information metric.
(slides 9) Visual Computing: Geometry, Graphics, and VisionFrank Nielsen
Those are the slides for the book:
Visual Computing: Geometry, Graphics, and Vision.
by Frank Nielsen (2005)
http://www.sonycsl.co.jp/person/nielsen/visualcomputing/
http://www.amazon.com/Visual-Computing-Geometry-Graphics-Charles/dp/1584504277
k-MLE: A fast algorithm for learning statistical mixture modelsFrank Nielsen
This document describes a fast algorithm called k-MLE for learning statistical mixture models. k-MLE is based on the connection between exponential family mixture models and Bregman divergences. It extends Lloyd's k-means clustering algorithm to optimize the complete log-likelihood of an exponential family mixture model using Bregman divergences. The algorithm iterates between assigning data points to clusters based on Bregman divergence, and updating the cluster parameters by taking the Bregman centroid of each cluster's assigned points. This provides a fast method for maximum likelihood estimation of exponential family mixture models.
This document provides an overview of computational geometry algorithms and their generalization to information spaces. It begins with a brief history of computational geometry and examples of libraries for geometric computing. The core concepts of Voronoi diagrams and their dual Delaunay complexes are reviewed for Euclidean spaces. These concepts are then generalized to Riemannian and dually affine computational information geometry, with applications to clustering and learning mixtures.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
Tailored Bregman Ball Trees for Effective Nearest NeighborsFrank Nielsen
This document presents an improved Bregman ball tree (BB-tree++) for efficient nearest neighbor search using Bregman divergences. The BB-tree++ speeds up construction using Bregman 2-means++ initialization and adapts the branching factor. It also handles symmetrized Bregman divergences and prioritizes closer nodes. Experiments on image retrieval with SIFT descriptors show the BB-tree++ outperforms the original BB-tree and random sampling, providing faster approximate nearest neighbor search.
INF442: Traitement des données massivesFrank Nielsen
French slides from curriculum INF442 at Ecole Polytechnique.
English book entitled "Introduction to HPC with MPI for Data Science" (Springer, 2016)
http://www.springer.com/us/book/9783319219028
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
This document discusses curved Mahalanobis distances in Cayley-Klein geometries and their application to classification. Specifically:
1. It introduces Mahalanobis distances and generalizes them to curved distances in Cayley-Klein geometries, which can model both elliptic and hyperbolic geometries.
2. It describes how to learn these curved Mahalanobis metrics using an adaptation of Large Margin Nearest Neighbors (LMNN) to the elliptic and hyperbolic cases.
3. Experimental results on several datasets show that curved Mahalanobis distances can achieve comparable or better classification accuracy than standard Mahalanobis distances.
On representing spherical videos (Frank Nielsen, CVPR 2001)Frank Nielsen
The document discusses different geometric shapes like cubes, dodecahedrons, and icosahedrons that can be used as envelopes. It also mentions that images can be interactively slid onto these envelopes and spherical maps can be generated using techniques like Buckyballer's unfolded icosahedron map or stratified random and Hammersley sequences.
Patch Matching with Polynomial Exponential Families and Projective DivergencesFrank Nielsen
This document presents a method called Polynomial Exponential Family-Patch Matching (PEF-PM) to solve the patch matching problem. PEF-PM models patch colors using polynomial exponential families (PEFs), which are universal smooth positive densities. It estimates PEFs using a Score Matching Estimator and accelerates batch estimation using Summed Area Tables. Patch similarity is measured using a statistical projective divergence called the symmetrized γ-divergence. Experiments show PEF-PM handles noise robustly, symmetries, and outperforms baseline methods.
This document summarizes Frank Nielsen's talk on divergence-based center clustering and their applications. Some key points:
- Center-based clustering aims to minimize an objective function that assigns data points to their closest cluster centers. This is an NP-hard problem when the number of dimensions and data points are greater than 1.
- Mixed divergences use dual centroids per cluster to define cluster assignments. Total Jensen divergences are proposed as a way to make divergences more robust by incorporating a conformal factor.
- For clustering when centroids do not have closed-form solutions, initialization methods like k-means++ can be used which randomly select initial seeds without computing centroids. Total Jensen k-means++
The dual geometry of Shannon informationFrank Nielsen
The document discusses the dual geometry of Shannon information. It covers:
1. Shannon entropy and related concepts like maximum entropy principle and exponential families.
2. The properties of Kullback-Leibler divergence including its interpretation as a statistical distance and relation to maximum entropy.
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Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
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* Live demos with code snippets
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A new implementation of k-MLE for mixture modelling of Wishart distributions
1. A new implementation of k-MLE for
mixture modelling of Wishart distributions
Christophe Saint-Jean Frank Nielsen
Geometric Science of Information 2013
August 28, 2013 - Mines Paris Tech
2. Application Context (1)
2/31
We are interested in clustering varying-length sets of multivariate
observations of same dim. p.
X1 =
0
@
3:6 0:05 4:
3:6 0:05 4:
3:6 0:05 4:
1
A; : : : ;XN =
0
BBBB@
5:3 0:5 2:5
3:6 0:5 3:5
1:6 0:5 4:6
1:6 0:5 5:1
2:9 0:5 6:1
1
CCCCA
Sample mean is a good but not discriminative enough feature.
Second order cross-product matrices tXiXi may capture some
relations between (column) variables.
3. Application Context (2)
3/31
The problem is now the clustering of a set of p p PSD matrices :
=
x1 = tX1X1; x2 = tX2X2; : : : ; xN = tXNXN
Examples of applications : multispectral/DTI/radar imaging,
motion retrieval system, ...
4. Application Context (2)
3/31
The problem is now the clustering of a set of p p PSD matrices :
=
x1 = tX1X1; x2 = tX2X2; : : : ; xN = tXNXN
Examples of applications : multispectral/DTI/radar imaging,
motion retrieval system, ...
5. Outline of this talk
4/31
1 MLE and Wishart Distribution
Exponential Family and Maximum Likehood Estimate
Wishart Distribution
Two sub-families of the Wishart Distribution
2 Mixture modeling with k-MLE
Original k-MLE
k-MLE for Wishart distributions
Heuristics for the initialization
3 Application to motion retrieval
6. Reminder : Exponential Family (EF)
5/31
An exponential family is a set of parametric probability distributions
EF = fp(x; ) = pF (x; ) = exp fht(x); i + k(x) F()j 2 g
Terminology:
source parameters.
natural parameters.
t(x) sucient statistic.
k(x) auxiliary carrier measure.
F() the log-normalizer:
dierentiable, strictly
convex
= f 2 RDjF() 1g
is an open convex set
Almost all commonly used distributions are EF members but
uniform, Cauchy distributions.
7. Reminder : Maximum Likehood Estimate (MLE)
6/31
Maximum Likehood Estimate principle is a very common
approach for
8. tting parameters of a distribution
^ = argmax
L(; ) = argmax
YN
i=1
p(xi ; ) = argmin
1
N
XN
i=1
log p(xi ; )
assuming a sample = fx1; x2; :::; xNg of i.i.d observations.
Log density have a convenient expression for EF members
log pF (x; ) = ht(x); i + k(x) F()
It follows
^ = argmax
XN
i=1
log pF (xi ; ) = argmax
h
XN
i=1
!
t(xi ); i NF()
9. MLE with EF
7/31
Since F is a strictly convex, dierentiable function, MLE
exists and is unique :
rF(^) =
1
N
XN
i=1
t(xi )
Ideally, we have a closed form :
^ = rF1
1
N
XN
i=1
!
t(xi )
Numerical methods including Newton-Raphson can be
successfully applied.
11. nition (Central Wishart distribution)
Wishart distribution characterizes empirical covariance matrices for
zero-mean gaussian samples:
Wd (X; n; S) =
jXj
nd1
2 exp
12
tr(S1X)
2
nd
2 jSj
n
2 d
n
2
where for x 0, d (x) =
d(d1)
4
Qd
j=1
x j1
2
is the
multivariate gamma function.
Remarks : n d 1, E[X] = nS
The multivariate generalization of the chi-square distribution.
12. Wishart Distribution as an EF
9/31
It's an exponential family:
logWd (X; n; S ) = n; log jXj R + S ;
1
2
X HS
+ k(X) F(n; S )
with k(X) = 0 and
(n; S ) = (
n d 1
2
; S1); t(X) = (log jXj;
1
2
X);
F(n; S ) =
n +
(d + 1)
2
(d log(2) log jS j)+log d
n +
(d + 1)
2
13. MLE for Wishart Distribution
10/31
In the case of the Wishart distribution, a closed form would be
obtained by solving the following system
^ = rF1
1
N
XN
i=1
!
t(xi )
8
:
d log(2) log jS j + d
n + (d+1)
2
= n
n + (d+1)
2
1
S = S
(1)
with n and S the expectation parameters and d the derivative
of the log d .
Unfortunately, no closed-form solution is known.
15. xed (n = 2n + d + 1)
Fn(S ) =
nd
2
log(2)
n
2
log jS j + log d
n
2
kn(X) =
n d 1
2
log jXj
Case S
16. xed (S = 1
S )
FS (n) =
n +
d + 1
2
log j2Sj + log d
n +
d + 1
2
kS (X) =
1
2
tr (S1X)
17. Two sub-families of the Wishart Distribution (2)
12/31
Both are exponential families and MLE equations are solvable !
Case n
18. xed:
n
2
^1
S =
1
N
XN
i=1
1
2
Xi =) ^S = Nn
XN
i=1
Xi
!1
(2)
Case S
19. xed :
^n = 1
d
1
N
XN
i=1
!
log jXi j log j2Sj
d + 1
2
; ^n 0 (3)
with 1
d the functional reciprocal of d .
20. An iterative estimator for the Wishart Distribution
13/31
Algorithm 1: An estimator for parameters of the Wishart
Input: A sample X1;X2; : : : ;XN of Sd
++
Output: Final values of ^n and ^S
Initialize ^n with some value 0;
repeat
Update ^S using Eq. 2 with n = 2^n + d + 1;
Update ^n using Eq. 3 with S the inverse matrix of ^S ;
until convergence of the likelihood;
21. Questions and open problems
14/31
From a sample of Wishart matrices, distr. parameters are
recovered in few iterations.
Major question : do you have a MLE ? probably ...
Minor question : sample size N = 1 ?
Under-determined system
Regularization by sampling around X1
23. nite) mixture is a
exible tool to model a more
complex distribution m:
m(x) =
Xk
j=1
wjpj (x); 0 wj 1;
Xk
j=1
wj = 1
where pj are the component distributions of the mixture, wj
the mixing proportions.
In our case, we consider pj as member of some parametric
family (EF)
m(x; ) =
Xk
j=1
wjpFj (x; j )
with = (w1;w2; :::;wk1; 1; 2; :::; k )
Expectation-Maximization is not fast enough [5] ...
24. Original k-MLE (primal form.) in one slide
16/31
Algorithm 2: k-MLE
Input: A sample = fx1; x2; :::; xNg, F1; F2; :::; Fk Bregman
generator
Output: Estimate ^
of mixture parameters
A good initialization for (see later);
repeat
repeat
foreach xi 2 do zi = argmaxj log w^jpFj (xi ; ^j );
foreach Cj := fxi 2 jzi = jg do ^j = MLEFj (Cj );
until Convergence of the complete likelihood;
Update mixing proportions : w^j = jCj j=N
until Further convergence of the complete likelihood;
25. k-MLE’s properties
17/31
Another formulation comes with the connection between EF
and Bregman divergences [3]:
log pF (x; ) = BF(t(x) : ) + F(t(x)) + k(x)
Bregman divergence BF (: : :) associated to a strictly convex
and dierentiable function F :
26. Original k-MLE (dual form.) in one slide
18/31
Algorithm 3: k-MLE
Input: A sample = fy1 = t(x1); y2 = x2; :::; yn = t(xN)g,
F
1 ; F
2 ; :::; F
k Bregman generator
Output: ^
= ( ^w1; ^w2; :::; ^wk1; ^1 = rF(^1); :::; ^k = rF(^k ))
A good initialization for (see later);
repeat
repeat
foreach xi 2 do zi = argminj
h
BF
j
(yi : ^j ) log w^j
i
;
foreach Cj := fxi 2 jzi = jg do ^j =
P
xi2Cj
yi=jCj j
until Convergence of the complete likelihood;
Update mixing proportions : w^j = jCj j=N
until Further convergence of the complete likelihood;
27. k-MLE for Wishart distributions
19/31
Practical considerations impose modi
28. cations of the algorithm:
During the assignment empty clusters may appear (High
dimensional data get this worse).
A possible solution is to consider Hartigan and Wang's
strategy [6] instead of Lloyd's strategy:
Optimally transfer one observation at a time
Update the parameters of involved clusters.
Stop when no transfer is possible.
This should guarantees non-empty clusters [7] but does not
work when considering weighted clusters...
Get back to an old school criterion : jCzi j 1
Experimentally shown to perform better in high dimension
than the Lloyd's strategy.
29. k-MLE - Hartigan and Wang
20/31
Criterion for potential transfer (Max):
log ^wzi pFzi
(xi ; ^zi )
log ^wz
i
pFz
i
(xi ; ^zi
)
1
i = argmaxj log w^jpFj (xi ; ^j )
with z
Update rules :
^zi = MLEFj (Czi nfxig)
^z
i
= MLEFj (Cz
i
[ fxig)
OR
Criterion for potential transfer (Min):
BF(yi : z
i
) log wz
i
BF(yi : zi ) log wzi
1
with z
i = argminj (BF(yi : j )
log wj )
Update rules :
zi =
jCzi jzi yi
jCzi j 1
z
i
=
jCz
i
jz
i
+ yi
jCz
i
j + 1
30. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
31. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
32. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
33. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
34. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
35. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
36. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
37. Towards a good initialization...
21/31
Classical initializations methods : random centers, random
partition, furthest point (2-approximation), ...
Better approach is k-means++ [8]:
Sampling prop. to sq. distance to the nearest center.
Fast and greedy approximation : (kN)
Probabilistic guarantee of good initialization:
OPTF k-meansF O(log k)OPTF
Dual Bregman divergence BF may replace the square distance
39. x k, the number of clusters
We propose on-the-
y cluster creation together with the
k-MLE++ (inspired by DP-k-means [9]) :
Create cluster when there exists observations contributing too
much to the loss function with already selected centers
41. x k, the number of clusters
We propose on-the-
y cluster creation together with the
k-MLE++ (inspired by DP-k-means [9]) :
Create cluster when there exists observations contributing too
much to the loss function with already selected centers
43. x k, the number of clusters
We propose on-the-
y cluster creation together with the
k-MLE++ (inspired by DP-k-means [9]) :
Create cluster when there exists observations contributing too
much to the loss function with already selected centers
It may overestimate the number of clusters...
44. Initialization with DP-k-MLE++
23/31
Algorithm 4: DP-k-MLE++
Input: A sample y1 = t(X1); : : : ; yN = t(XN), F , 0
Output: C a subset of y1; : : : ; yN, k the number of clusters
Choose
45. rst seed C = fyjg, for j uniformly random in f1; 2; : : : ;Ng;
repeat
foreach yi do compute pi = BF(yi : C)=
PN
i 0=1 BF(yi 0 : C)
where BF(yi : C) = minc2CBF(yi : c) ;
if 9pi then
Choose next seed s among y1; y2; : : : ; yN with prob. pi ;
Add selected seed to C : C = C [ fsg ;
until all pi ;
k = jCj;
46. Motion capture
24/31
Real dataset:
Motion capture of contemporary dancers (15 sensors in 3d).
47. Application to motion retrieval(1)
25/31
Motion capture data can be view as matrices Xi with dierent
row sizes but same column size d.
The idea is to describe Xi through one mixture model
parameters ^
i .
48. Application to motion retrieval(1)
25/31
Motion capture data can be view as matrices Xi with dierent
row sizes but same column size d.
The idea is to describe Xi through one mixture model
parameters ^
i .
49. Application to motion retrieval(1)
25/31
Motion capture data can be view as matrices Xi with dierent
row sizes but same column size d.
The idea is to describe Xi through one mixture model
parameters ^
i .
50. Application to motion retrieval(1)
25/31
Motion capture data can be view as matrices Xi with dierent
row sizes but same column size d.
The idea is to describe Xi through one mixture model
parameters ^
i .
51. Application to motion retrieval(1)
25/31
Motion capture data can be view as matrices Xi with dierent
row sizes but same column size d.
The idea is to describe Xi through one mixture model
parameters ^
i .
Remark: Size of each sub-motion is known (so its n)
52. Application to motion retrieval(1)
25/31
Motion capture data can be view as matrices Xi with dierent
row sizes but same column size d.
The idea is to describe Xi through one mixture model
parameters ^
i .
Mixture parameters can be viewed as a sparse representation
of local dynamics in Xi .
53. Application to motion retrieval(2)
26/31
Comparing two movements amounts to compute a
dissimilarity measure between ^
i and ^
j .
Remark 1 : with DP-k-MLE++, the two mixtures would not
probably have the same number of components.
Remark 2 : when both mixtures have one component, a
natural choice is
KL(Wd (:; ^)jjWd (:; ^0)) = BF(^ : ^0) = BF (^0 : ^)
A closed form is always available !
No closed form exists for KL divergence between general
mixtures.
54. Application to motion retrieval(3)
27/31
A possible solution is to use the CS divergence [10]:
CS(m : m0) = log
R
m(x)m0 R (x)dx
m(x)2dx
R
m0(x)2dx
It has a analytic formula for
Z
m(x)m0(x)dx =
Xk
j=1
k0 X
j 0=1
j 0 expF(j+0
wjw0
j0 )(F(j)+F(0
j0 ))
+
Note that this expression is well de
57. c code in MatlabTM.
Today implementation in Python (based on pyMEF [2])
Ongoing proof of concept (with Herranz F., Beurive A.)
58. Conclusions - Future works
29/31
Still some mathematical work to be done:
Solve MLE equations to get rF = (rF)1 then F
Characterize our estimator for full Wishart distribution.
Complete and validate the prototype of system for motion
retrieval.
Speeding-up algorithm: computational/numerical/algorithmic
tricks.
library for bregman divergences learning ?
Possible extensions:
Reintroduce mean vector in the model : Gaussian-Wishart
Online k-means - online k-MLE ...
59. References I
30/31
Nielsen, F.:
k-MLE: A fast algorithm for learning statistical mixture models.
In: International Conference on Acoustics, Speech and Signal Processing.
(2012) pp. 869{872
Schwander, O. and Nielsen, F.
pyMEF - A framework for Exponential Families in Python
in Proceedings of the 2011 IEEE Workshop on Statistical Signal Processing
Banerjee, A., Merugu, S., Dhillon, I.S., Ghosh, J.
Clustering with bregman divergences.
Journal of Machine Learning Research (6) (2005) 1705{1749
Nielsen, F., Garcia, V.:
Statistical exponential families: A digest with
ash cards.
http://arxiv.org/abs/0911.4863 (11 2009)
Hidot, S., Saint Jean, C.:
An Expectation-Maximization algorithm for the Wishart mixture model:
Application to movement clustering.
Pattern Recognition Letters 31(14) (2010) 2318{2324
60. References II
31/31
Hartigan, J.A., Wong, M.A.:
Algorithm AS 136: A k-means clustering algorithm.
Journal of the Royal Statistical Society. Series C (Applied Statistics) 28(1)
(1979) 100{108
Telgarsky, M., Vattani, A.:
Hartigan's method: k-means clustering without Voronoi.
In: Proc. of International Conference on Arti
61. cial Intelligence and
Statistics (AISTATS). (2010) pp. 820{827
Arthur, D., Vassilvitskii, S.:
k-means++: The advantages of careful seeding
In: Proceedings of the eighteenth annual ACM-SIAM symposium on
Discrete algorithms (2007) pp. 1027{1035
Kulis, B., Jordan, M.I.:
Revisiting k-means: New algorithms via Bayesian nonparametrics.
In: International Conference on Machine Learning (ICML). (2012)
Nielsen, F.:
Closed-form information-theoretic divergences for statistical mixtures.
In: International Conference on Pattern Recognition (ICPR). (2012) pp.
1723{1726