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48
Clustering with Bregman Divergences
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2005
"... A wide variety of distortion functions are used for clustering, e.g., squared Euclidean distance, Mahalanobis distance and relative entropy. In this paper, we propose and analyze parametric hard and soft clustering algorithms based on a large class of distortion functions known as Bregman divergence ..."
Abstract

Cited by 310 (51 self)
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A wide variety of distortion functions are used for clustering, e.g., squared Euclidean distance, Mahalanobis distance and relative entropy. In this paper, we propose and analyze parametric hard and soft clustering algorithms based on a large class of distortion functions known as Bregman divergences. The proposed algorithms unify centroidbased parametric clustering approaches, such as classical kmeans and informationtheoretic clustering, which arise by special choices of the Bregman divergence. The algorithms maintain the simplicity and scalability of the classical kmeans algorithm, while generalizing the basic idea to a very large class of clustering loss functions. There are two main contributions in this paper. First, we pose the hard clustering problem in terms of minimizing the loss in Bregman information, a quantity motivated by ratedistortion theory, and present an algorithm to minimize this loss. Secondly, we show an explicit bijection between Bregman divergences and exponential families. The bijection enables the development of an alternative interpretation of an ecient EM scheme for learning models involving mixtures of exponential distributions. This leads to a simple soft clustering algorithm for all Bregman divergences.
Modelling Dependent Defaults
 RISK
, 2000
"... We consider the modelling of dependent defaults using latent variable models (the approach that underlies KMV and CreditMetrics) and mixture models (the approach underlying CreditRisk+). We explore the role of copulas in the latent variable framework and present results from a simulation study sh ..."
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Cited by 31 (6 self)
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We consider the modelling of dependent defaults using latent variable models (the approach that underlies KMV and CreditMetrics) and mixture models (the approach underlying CreditRisk+). We explore the role of copulas in the latent variable framework and present results from a simulation study showing that even for fixed asset correlation assumptions concerning the dependence of the latent variables can have a large effect on the distribution of credit losses. We explore the effect of the tail of the mixingdistribution for the tail of the creditloss distributions. Finally, we discuss the relation between latent variable models and mixture models and provide general conditions under which these models can be mapped into each other. Our contribution can be viewed as an analysis of the model risk associated with the modelling of dependence between credit losses.
Asymptotic behavior of an allocation policy for revenue management
 Oper. Res
"... Revenue management has become an important tool in the airline, hotel, and rental car industries. We describe asymptotic properties of revenue management policies derived from the solution of a deterministic optimization problem. Our primary results state that, within a stochastic and dynamic framew ..."
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Cited by 21 (4 self)
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Revenue management has become an important tool in the airline, hotel, and rental car industries. We describe asymptotic properties of revenue management policies derived from the solution of a deterministic optimization problem. Our primary results state that, within a stochastic and dynamic framework, solutions arising out of a single wellknown linear program can be used to generate allocation policies for which the normalized revenue converges in distribution to a constant upper bound on the optimal value. We also show similar asymptotic results for expected revenues. In addition, we describe counterintuitive behavior that can occur when allocations are updated during the booking process (updating allocations can lead to lower expected revenue). These results add to the understanding of allocation policies and help to make concrete the statement that simple policies from easytosolve formulations can be relatively effective, even when analyzed in the more realistic stochastic and dynamic framework. 1.
Ordering Monte Carlo Markov Chains
 School of Statistics, University of Minnesota
, 1999
"... Markov chains having the same stationary distribution ß can be partially ordered by performance in the central limit theorem. We say that one chain is at least as good as another in the efficiency partial ordering if the variance in the central limit theorem is at least as small for every L 2 (ß) ..."
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Cited by 21 (5 self)
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Markov chains having the same stationary distribution ß can be partially ordered by performance in the central limit theorem. We say that one chain is at least as good as another in the efficiency partial ordering if the variance in the central limit theorem is at least as small for every L 2 (ß) functional of the chain. Peskun partial ordering implies efficiency partial ordering [25, 30]. Here we show that Peskun partial ordering implies, for finite state spaces, ordering of all the eigenvalues of the transition matrices, and, for general state spaces, ordering of the suprema of the spectra of the transition operators. We also define a covariance partial ordering based on lag one autocovariances and show that it is equivalent to the efficiency partial ordering when restricted to reversible Markov chains. Similar but weaker results are provided for nonreversible Markov chains. Keywords: Peskun ordering, Eigenvalues, Spectral decomposition, Nonreversible kernels. 1 Introduction I...
Ordering, Slicing And Splitting Monte Carlo Markov Chains
, 1998
"... Markov chain Monte Carlo is a method of approximating the integral of a function f with respect to a distribution ß. A Markov chain that has ß as its stationary distribution is simulated producing samples X 1 ; X 2 ; : : : . The integral is approximated by taking the average of f(X n ) over the sam ..."
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Cited by 9 (3 self)
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Markov chain Monte Carlo is a method of approximating the integral of a function f with respect to a distribution ß. A Markov chain that has ß as its stationary distribution is simulated producing samples X 1 ; X 2 ; : : : . The integral is approximated by taking the average of f(X n ) over the sample path. The standard way to construct such Markov chains is the MetropolisHastings algorithm. The class P of all Markov chains having ß as their unique stationary distribution is very large, so it is important to have criteria telling when one chain performs better than another. The Peskun ordering is a partial ordering on P. If two Markov chains are Peskun ordered, then the better chain has smaller variance in the central limit theorem for every function f that has a variance. Peskun ordering is sufficient for this but not necessary. We study the implications of the Peskun ordering both in finite and general state spaces. Unfortunately there are many MetropolisHastings samplers that are...
Applications of the continuoustime ballot theorem to Brownian motion and related processes
, 2001
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Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices
 Ann. Stat
, 2011
"... Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n×p random matr ..."
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Cited by 8 (4 self)
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Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n×p random matrix in the highdimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.