TY - JOUR
T1 - Clustering with Bregman divergences
AU - Banerjee, Arindam
AU - Merugu, Srujana
AU - Dhillon, Inderjit S.
AU - Ghosh, Joydeep
PY - 2005/10
Y1 - 2005/10
N2 - A wide variety of distortion functions, such as squared Euclidean distance, Mahalanobis distance, Itakura-Saito distance and relative entropy, have been used for clustering. 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 centroid-based parametric clustering approaches, such as classical kmeans, the Linde-Buzo-Gray (LBG) algorithm and information-theoretic 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 method to a large class of clustering loss functions. This is achieved by first posing the hard clustering problem in terms of minimizing the loss in Bregman information, a quantity motivated by rate distortion theory, and then deriving an iterative algorithm that monotonically decreases this loss. In addition, we show that there is a bijection between regular exponential families and a large class of Bregman divergences, that we call regular Bregman divergences. This result enables the development of an alternative interpretation of an efficient EM scheme for learning mixtures of exponential family distributions, and leads to a simple soft clustering algorithm for regular Bregman divergences. Finally, we discuss the connection between rate distortion theory and Bregman clustering and present an information theoretic analysis of Bregman clustering algorithms in terms of a trade-off between compression and loss in Bregman information.
AB - A wide variety of distortion functions, such as squared Euclidean distance, Mahalanobis distance, Itakura-Saito distance and relative entropy, have been used for clustering. 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 centroid-based parametric clustering approaches, such as classical kmeans, the Linde-Buzo-Gray (LBG) algorithm and information-theoretic 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 method to a large class of clustering loss functions. This is achieved by first posing the hard clustering problem in terms of minimizing the loss in Bregman information, a quantity motivated by rate distortion theory, and then deriving an iterative algorithm that monotonically decreases this loss. In addition, we show that there is a bijection between regular exponential families and a large class of Bregman divergences, that we call regular Bregman divergences. This result enables the development of an alternative interpretation of an efficient EM scheme for learning mixtures of exponential family distributions, and leads to a simple soft clustering algorithm for regular Bregman divergences. Finally, we discuss the connection between rate distortion theory and Bregman clustering and present an information theoretic analysis of Bregman clustering algorithms in terms of a trade-off between compression and loss in Bregman information.
KW - Bregman divergences
KW - Bregman information
KW - Clustering
KW - Expectation maximization
KW - Exponential families
KW - Information theory
UR - http://www.scopus.com/inward/record.url?scp=26244461684&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=26244461684&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:26244461684
SN - 1533-7928
VL - 6
JO - Journal of Machine Learning Research
JF - Journal of Machine Learning Research
ER -