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Duality and Auxiliary Functions for Bregman Distances
 SCHOOL OF COMPUTER SCIENCE, CARNEGIE MELLON UNIVERSITY
, 2002
"... We formulate and prove a convex duality theorem for Bregman distances and present a technique based on auxiliary functions for deriving and proving convergence of iterative algorithms to minimize Bregman distance subject to linear constraints. ..."
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Cited by 29 (1 self)
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We formulate and prove a convex duality theorem for Bregman distances and present a technique based on auxiliary functions for deriving and proving convergence of iterative algorithms to minimize Bregman distance subject to linear constraints.
Bregman distances and Chebyshev sets
, 2007
"... A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinitedimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the ..."
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Cited by 5 (1 self)
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if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory
Logistic Regression, AdaBoost and Bregman Distances
, 2000
"... We give a unified account of boosting and logistic regression in which each learning problem is cast in terms of optimization of Bregman distances. The striking similarity of the two problems in this framework allows us to design and analyze algorithms for both simultaneously, and to easily adapt al ..."
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Cited by 259 (45 self)
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We give a unified account of boosting and logistic regression in which each learning problem is cast in terms of optimization of Bregman distances. The striking similarity of the two problems in this framework allows us to design and analyze algorithms for both simultaneously, and to easily adapt
Bregman distances and Klee sets
, 2008
"... In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee se ..."
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Cited by 4 (1 self)
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sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then — analogously to the Euclidean distance case — every
Statistical Learning Algorithms Based on Bregman Distances
, 1997
"... We present a class of statistical learning algorithms formulated in terms of minimizing Bregman distances, a family of generalized entropy measures associated with convex functions. The inductive learning scheme is akin to growing a decision tree, with the Bregman distance filling the role of the im ..."
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Cited by 30 (1 self)
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We present a class of statistical learning algorithms formulated in terms of minimizing Bregman distances, a family of generalized entropy measures associated with convex functions. The inductive learning scheme is akin to growing a decision tree, with the Bregman distance filling the role
Statistical Learning Algorithms Based on Bregman Distances
"... Abstract  We present a class of statistical learning algorithms formulated in terms of minimizing Bregman distances, a family of generalized entropy measures associated with convex functions. The inductive learning scheme is akin to growing a decision tree, with the Bregman distance lling the role ..."
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Abstract  We present a class of statistical learning algorithms formulated in terms of minimizing Bregman distances, a family of generalized entropy measures associated with convex functions. The inductive learning scheme is akin to growing a decision tree, with the Bregman distance lling the role
On bregman distances and divergences of probability measures
 IEEE Trans. on Information Theory
"... iv ..."
On Bregman Distances and Divergences of 1 Probability Measures
"... The paper introduces scaled Bregman distances of probability distributions which admit nonuniform contributions of observed events. They are introduced in a general form covering not only the distances of discrete and continuous stochastic observations, but also the distances of random processes an ..."
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The paper introduces scaled Bregman distances of probability distributions which admit nonuniform contributions of observed events. They are introduced in a general form covering not only the distances of discrete and continuous stochastic observations, but also the distances of random processes
Klee sets and Chebyshev centers for the right Bregman distance
, 2010
"... We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function is spe ..."
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We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function
Bregman Distance to L1 Regularized Logistic Regression
"... In this work we investigate the relationship between Bregman distances and regularized Logistic Regression model. We convert L1regularized logistic regression (LR) into more general Bregman divergence framework and propose a primaldual method based algorithm for learning the parameters of the mode ..."
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In this work we investigate the relationship between Bregman distances and regularized Logistic Regression model. We convert L1regularized logistic regression (LR) into more general Bregman divergence framework and propose a primaldual method based algorithm for learning the parameters
Results 1  10
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