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3,653
Xmeans: Extending Kmeans with Efficient Estimation of the Number of Clusters
 In Proceedings of the 17th International Conf. on Machine Learning
, 2000
"... Despite its popularity for general clustering, Kmeans suffers three major shortcomings; it scales poorly computationally, the number of clusters K has to be supplied by the user, and the search is prone to local minima. We propose solutions for the first two problems, and a partial remedy for the t ..."
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Cited by 267 (5 self)
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Despite its popularity for general clustering, Kmeans suffers three major shortcomings; it scales poorly computationally, the number of clusters K has to be supplied by the user, and the search is prone to local minima. We propose solutions for the first two problems, and a partial remedy for the third. Building on prior work for algorithmic acceleration that is not based on approximation, we introduce a new algorithm that efficiently, searches the space of cluster locations and number of clusters to optimize the Bayesian Information Criterion (BIC) or the Akaike Information Criterion (AIC) measure. The innovations include two new ways of exploiting cached sufficient statistics and a new very efficient test that in one Kmeans sweep selects the most promising subset of classes for refinement. This gives rise to a fast, statistically founded algorithm that outputs both the number of classes and their parameters. Experiments show this technique reveals the true number of classes in the underlying distribution, and that it is much faster than repeatedly using accelerated Kmeans for different values of K.
A Unifying Review of Linear Gaussian Models
, 1999
"... Factor analysis, principal component analysis, mixtures of gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models can all be unified as variations of unsupervised learning under a single basic generative model. This is achieved by collecting together disparate observa ..."
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Cited by 260 (17 self)
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Factor analysis, principal component analysis, mixtures of gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models can all be unified as variations of unsupervised learning under a single basic generative model. This is achieved by collecting together disparate observations and derivations made by many previous authors and introducing a new way of linking discrete and continuous state models using a simple nonlinearity. Through the use of other nonlinearities, we show how independent component analysis is also a variation of the same basic generative model. We show that factor analysis and mixtures of gaussians can be implemented in autoencoder neural networks and learned using squared error plus the same regularization term. We introduce a new model for static data, known as sensible principal component analysis, as well as a novel concept of spatially adaptive observation noise. We also review some of the literature involving global and local mixtures of the basic models and provide pseudocode for inference and learning for all the basic models.
Soft Margins for AdaBoost
, 1998
"... Recently ensemble methods like AdaBoost were successfully applied to character recognition tasks, seemingly defying the problems of overfitting. This paper shows that although AdaBoost rarely overfits in the low noise regime it clearly does so for higher noise levels. Central for understanding this ..."
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Cited by 256 (22 self)
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Recently ensemble methods like AdaBoost were successfully applied to character recognition tasks, seemingly defying the problems of overfitting. This paper shows that although AdaBoost rarely overfits in the low noise regime it clearly does so for higher noise levels. Central for understanding this fact is the margin distribution and we find that AdaBoost achieves  doing gradient descent in an error function with respect to the margin  asymptotically a hard margin distribution, i.e. the algorithm concentrates its resources on a few hardtolearn patterns (here an interesting overlap emerge to Support Vectors). This is clearly a suboptimal strategy in the noisy case, and regularization, i.e. a mistrust in the data, must be introduced in the algorithm to alleviate the distortions that a difficult pattern (e.g. outliers) can cause to the margin distribution. We propose several regularization methods and generalizations of the original AdaBoost algorithm to achieve a soft margin  a ...
Benchmarking Least Squares Support Vector Machine Classifiers
 Neural Processing Letters
"... In Support Vector Machines (SVMs), the solution of the classification problem is characterized by a (convex) quadratic programming (QP) problem. In a modified version of SVMs, called Least Squares SVM classifiers (LSSVMs), a least squares cost function is proposed so as to obtain a linear set of eq ..."
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Cited by 253 (37 self)
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In Support Vector Machines (SVMs), the solution of the classification problem is characterized by a (convex) quadratic programming (QP) problem. In a modified version of SVMs, called Least Squares SVM classifiers (LSSVMs), a least squares cost function is proposed so as to obtain a linear set of equations in the dual space. While the SVM classifier has a large margin interpretation, the LSSVM formulation is related in this paper to a ridge regression approach for classification with binary targets and to Fisher's linear discriminant analysis in the feature space. Multiclass categorization problems are represented by a set of binary classifiers using different output coding schemes. While regularization is used to control the effective number of parameters of the LSSVM classifier, the sparseness property of SVMs is lost due to the choice of the 2norm. Sparseness can be imposed in a second stage by gradually pruning the support value spectrum and optimizing the hyperparameters during the sparse approximation procedure. In this paper, twenty public domain benchmark datasets are used to evaluate the test set performance of LSSVM classifiers with linear, polynomial and radial basis function (RBF) kernels. Both the SVM and LSSVM classifier with RBF kernel in combination with standard crossvalidation procedures for hyperparameter selection achieve comparable test set performances. These SVM and LSSVM performances are consistently very good when compared to a variety of methods described in the literature including decision tree based algorithms, statistical algorithms and instance based learning methods. We show on ten UCI datasets that the LSSVM sparse approximation procedure can be successfully applied.
Segmentation of brain MR images through a hidden Markov random field model and the expectationmaximization algorithm
 IEEE Transactions on Medical. Imaging
, 2001
"... Abstract—The finite mixture (FM) model is the most commonly used model for statistical segmentation of brain magnetic resonance (MR) images because of its simple mathematical form and the piecewise constant nature of ideal brain MR images. However, being a histogrambased model, the FM has an intrin ..."
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Cited by 250 (10 self)
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Abstract—The finite mixture (FM) model is the most commonly used model for statistical segmentation of brain magnetic resonance (MR) images because of its simple mathematical form and the piecewise constant nature of ideal brain MR images. However, being a histogrambased model, the FM has an intrinsic limitation—no spatial information is taken into account. This causes the FM model to work only on welldefined images with low levels of noise; unfortunately, this is often not the the case due to artifacts such as partial volume effect and bias field distortion. Under these conditions, FM modelbased methods produce unreliable results. In this paper, we propose a novel hidden Markov random field (HMRF) model, which is a stochastic process generated by a MRF whose state sequence cannot be observed directly but which can be indirectly estimated through observations. Mathematically, it can be shown that the FM model is a degenerate version of the HMRF model. The advantage of the HMRF model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites. Although MRF modeling has been employed in MR image segmentation by other researchers, most reported methods are limited to using MRF as a general prior in an FM modelbased approach. To fit the HMRF model, an EM algorithm is used. We show that by incorporating both the HMRF model and the EM algorithm into a HMRFEM framework, an accurate and robust segmentation can be achieved. More importantly, the HMRFEM framework can easily be combined with other techniques. As an example, we show how the bias field correction algorithm of Guillemaud and Brady (1997) can be incorporated into this framework to achieve a threedimensional fully automated approach for brain MR image segmentation. Index Terms—Bias field correction, expectationmaximization, hidden Markov random field, MRI, segmentation. I.
Scaling Clustering Algorithms to Large Databases”, Microsoft Research Report
, 1998
"... Practical clustering algorithms require multiple data scans to achieve convergence. For large databases, these scans become prohibitively expensive. We present a scalable clustering framework applicable to a wide class of iterative clustering. We require at most one scan of the database. In this wor ..."
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Cited by 244 (5 self)
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Practical clustering algorithms require multiple data scans to achieve convergence. For large databases, these scans become prohibitively expensive. We present a scalable clustering framework applicable to a wide class of iterative clustering. We require at most one scan of the database. In this work, the framework is instantiated and numerically justified with the popular KMeans clustering algorithm. The method is based on identifying regions of the data that are compressible, regions that must be maintained in memory, and regions that are discardable. The algorithm operates within the confines of a limited memory buffer. Empirical results demonstrate that the scalable scheme outperforms a samplingbased approach. In our scheme, data resolution is preserved to the extent possible based upon the size of the allocated memory buffer and the fit of current clustering model to the data. The framework is naturally extended to update multiple clustering models simultaneously. We empirically evaluate on synthetic and publicly available data sets.
Refining Initial Points for KMeans Clustering
, 1998
"... Practical approaches to clustering use an iterative procedure (e.g. KMeans, EM) which converges to one of numerous local minima. It is known that these iterative techniques are especially sensitive to initial starting conditions. We present a procedure for computing a refined starting condition fro ..."
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Cited by 233 (5 self)
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Practical approaches to clustering use an iterative procedure (e.g. KMeans, EM) which converges to one of numerous local minima. It is known that these iterative techniques are especially sensitive to initial starting conditions. We present a procedure for computing a refined starting condition from a given initial one that is based on an efficient technique for estimating the modes of a distribution. The refined initial starting condition allows the iterative algorithm to converge to a "better" local minimum. The procedure is applicable to a wide class of clustering algorithms for both discrete and continuous data. We demonstrate the application of this method to the popular KMeans clustering algorithm and show that refined initial starting points indeed lead to improved solutions. Refinement run time is considerably lower than the time required to cluster the full database. The method is scalable and can be coupled with a scalable clustering algorithm to address the largescale cl...
Survey of clustering algorithms
 IEEE TRANSACTIONS ON NEURAL NETWORKS
, 2005
"... Data analysis plays an indispensable role for understanding various phenomena. Cluster analysis, primitive exploration with little or no prior knowledge, consists of research developed across a wide variety of communities. The diversity, on one hand, equips us with many tools. On the other hand, the ..."
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Cited by 231 (3 self)
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Data analysis plays an indispensable role for understanding various phenomena. Cluster analysis, primitive exploration with little or no prior knowledge, consists of research developed across a wide variety of communities. The diversity, on one hand, equips us with many tools. On the other hand, the profusion of options causes confusion. We survey clustering algorithms for data sets appearing in statistics, computer science, and machine learning, and illustrate their applications in some benchmark data sets, the traveling salesman problem, and bioinformatics, a new field attracting intensive efforts. Several tightly related topics, proximity measure, and cluster validation, are also discussed.
Online selection of discriminative tracking features
, 2003
"... This paper presents an online feature selection mechanism for evaluating multiple features while tracking and adjusting the set of features used to improve tracking performance. Our hypothesis is that the features that best discriminate between object and background are also best for tracking the ..."
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Cited by 224 (5 self)
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This paper presents an online feature selection mechanism for evaluating multiple features while tracking and adjusting the set of features used to improve tracking performance. Our hypothesis is that the features that best discriminate between object and background are also best for tracking the object. Given a set of seed features, we compute log likelihood ratios of class conditional sample densities from object and background to form a new set of candidate features tailored to the local object/background discrimination task. The twoclass variance ratio is used to rank these new features according to how well they separate sample distributions of object and background pixels. This feature evaluation mechanism is embedded in a meanshift tracking system that adaptively selects the topranked discriminative features for tracking. Examples are presented that demonstrate how this method adapts to changing appearances of both tracked object and scene background. We note susceptibility of the variance ratio feature selection method to distraction by spatially correlated background clutter, and develop an additional approach that seeks to minimize the likelihood of distraction.
On the mathematical foundations of learning
 Bulletin of the American Mathematical Society
, 2002
"... The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton ..."
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Cited by 223 (12 self)
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The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton