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231
An introduction to kernelbased learning algorithms
 IEEE TRANSACTIONS ON NEURAL NETWORKS
, 2001
"... This paper provides an introduction to support vector machines (SVMs), kernel Fisher discriminant analysis, and ..."
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Cited by 598 (55 self)
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This paper provides an introduction to support vector machines (SVMs), kernel Fisher discriminant analysis, and
On the distribution of the largest eigenvalue in principal components analysis
 ANN. STATIST
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
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Cited by 422 (4 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.
Generalized Discriminant Analysis Using a Kernel Approach
, 2000
"... We present a new method that we call Generalized Discriminant Analysis (GDA) to deal with nonlinear discriminant analysis using kernel function operator. The underlying theory is close to the Support Vector Machines (SVM) insofar as the GDA method provides a mapping of the input vectors into high di ..."
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Cited by 336 (2 self)
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We present a new method that we call Generalized Discriminant Analysis (GDA) to deal with nonlinear discriminant analysis using kernel function operator. The underlying theory is close to the Support Vector Machines (SVM) insofar as the GDA method provides a mapping of the input vectors into high dimensional feature space. In the transformed space, linear properties make it easy to extend and generalize the classical Linear Discriminant Analysis (LDA) to non linear discriminant analysis. The formulation is expressed as an eigenvalue problem resolution. Using a different kernel, one can cover a wide class of nonlinearities. For both simulated data and alternate kernels, we give classification results as well as the shape of the separating function. The results are confirmed using a real data to perform seed classification. 1. Introduction Linear discriminant analysis (LDA) is a traditional statistical method which has proven successful on classification problems [Fukunaga, 1990]. The p...
Shape quantization and recognition with randomized trees
 NEURAL COMPUTATION
, 1997
"... We explore a new approach to shape recognition based on a virtually infinite family of binary features ("queries") of the image data, designed to accommodate prior information about shape invariance and regularity. Each query corresponds to a spatial arrangement ofseveral local topographic ..."
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Cited by 263 (18 self)
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We explore a new approach to shape recognition based on a virtually infinite family of binary features ("queries") of the image data, designed to accommodate prior information about shape invariance and regularity. Each query corresponds to a spatial arrangement ofseveral local topographic codes ("tags") which are in themselves too primitive and common to be informative about shape. All the discriminating power derives from relative angles and distances among the tags. The important attributes of the queries are (i) a natural partial ordering corresponding to increasing structure and complexity; (ii) semiinvariance, meaning that most shapes of a given class will answer the same way to two queries which are successive in the ordering; and (iii) stability, since the queries are not based on distinguished points and substructures. No classifier based on the full feature set can be evaluated and it is impossible to determine a priori which arrangements are informative. Our approach is to select informative features and build tree classifiers at the same time by inductive learning. In effect, each tree provides an approximation to the full posterior where the features
A Comparison of Prediction Accuracy, Complexity, and Training Time of Thirtythree Old and New Classification Algorithms
, 2000
"... . Twentytwo decision tree, nine statistical, and two neural network algorithms are compared on thirtytwo datasets in terms of classication accuracy, training time, and (in the case of trees) number of leaves. Classication accuracy is measured by mean error rate and mean rank of error rate. Both cr ..."
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Cited by 234 (8 self)
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. Twentytwo decision tree, nine statistical, and two neural network algorithms are compared on thirtytwo datasets in terms of classication accuracy, training time, and (in the case of trees) number of leaves. Classication accuracy is measured by mean error rate and mean rank of error rate. Both criteria place a statistical, splinebased, algorithm called Polyclass at the top, although it is not statistically signicantly dierent from twenty other algorithms. Another statistical algorithm, logistic regression, is second with respect to the two accuracy criteria. The most accurate decision tree algorithm is Quest with linear splits, which ranks fourth and fth, respectively. Although splinebased statistical algorithms tend to have good accuracy, they also require relatively long training times. Polyclass, for example, is third last in terms of median training time. It often requires hours of training compared to seconds for other algorithms. The Quest and logistic regression algor...
Support vector machines: Training and applications
 A.I. MEMO 1602, MIT A. I. LAB
, 1997
"... The Support Vector Machine (SVM) is a new and very promising classification technique developed by Vapnik and his group at AT&T Bell Laboratories [3, 6, 8, 24]. This new learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and MultiLayer Perc ..."
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Cited by 223 (3 self)
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The Support Vector Machine (SVM) is a new and very promising classification technique developed by Vapnik and his group at AT&T Bell Laboratories [3, 6, 8, 24]. This new learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and MultiLayer Perceptron classifiers. The main idea behind the technique is to separate the classes with a surface that maximizes the margin between them. An interesting property of this approach is that it is an approximate implementation of the Structural Risk Minimization (SRM) induction principle [23]. The derivation of Support Vector Machines, its relationship with SRM, and its geometrical insight, are discussed in this paper. Since Structural Risk Minimization is an inductive principle that aims at minimizing a bound on the generalization error of a model, rather than minimizing the Mean Square Error over the data set (as Empirical Risk Minimization methods do), training a SVM to obtain the maximum margin classi er requires a different objective function. This objective function is then optimized by solving a largescale quadratic programming problem with linear and box constraints. The problem is considered challenging, because the quadratic form is completely dense, so the memory
Discriminant Analysis by Gaussian Mixtures
 Journal of the Royal Statistical Society, Series B
, 1996
"... FisherRao linear discriminant analysis (LDA) is a valuable tool for multigroup classification. LDA is equivalent to maximum likelihood classification assuming Gaussian distributions for each class. In this paper, we fit Gaussian mixtures to each class to facilitate effective classification in nonn ..."
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Cited by 214 (9 self)
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FisherRao linear discriminant analysis (LDA) is a valuable tool for multigroup classification. LDA is equivalent to maximum likelihood classification assuming Gaussian distributions for each class. In this paper, we fit Gaussian mixtures to each class to facilitate effective classification in nonnormal settings, especially when the classes are clustered. Low dimensional views are an important byproduct of LDAour new techniques inherit this feature. We are able to control the withinclass spread of the subclass centers relative to the betweenclass spread. Our technique for fitting these models permits a natural blend with nonparametric versions of LDA. Keywords: Classification, Pattern Recognition, Clustering, Nonparametric, Penalized. 1 Introduction In the generic classification or discrimination problem, the outcome of interest G falls into J unordered classes, which for convenience we denote by the set J = f1; 2; 3; \Delta \Delta \Delta Jg. We wish to build a rule for pred...
Model Selection and the Principle of Minimum Description Length
 Journal of the American Statistical Association
, 1998
"... This paper reviews the principle of Minimum Description Length (MDL) for problems of model selection. By viewing statistical modeling as a means of generating descriptions of observed data, the MDL framework discriminates between competing models based on the complexity of each description. This ..."
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Cited by 200 (8 self)
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This paper reviews the principle of Minimum Description Length (MDL) for problems of model selection. By viewing statistical modeling as a means of generating descriptions of observed data, the MDL framework discriminates between competing models based on the complexity of each description. This approach began with Kolmogorov's theory of algorithmic complexity, matured in the literature on information theory, and has recently received renewed interest within the statistics community. In the pages that follow, we review both the practical as well as the theoretical aspects of MDL as a tool for model selection, emphasizing the rich connections between information theory and statistics. At the boundary between these two disciplines, we find many interesting interpretations of popular frequentist and Bayesian procedures. As we will see, MDL provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared. We illustrate th...
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
, 2008
"... ..."
Eigenvalues of large sample covariance matrices of spiked population models
, 2006
"... We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the nonunit population ones when both sample size and population size become large. This pape ..."
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Cited by 163 (8 self)
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We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the nonunit population ones when both sample size and population size become large. This paper completely determines the almost sure limits for a general class of samples. 1