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18
Sided and symmetrized Bregman centroids
 IEEE Transactions on Information Theory
, 2009
"... Abstract—We generalize the notions of centroids (and barycenters) to the broad class of informationtheoretic distortion measures called Bregman divergences. Bregman divergences form a rich and versatile family of distances that unifies quadratic Euclidean distances with various wellknown statistic ..."
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Cited by 37 (13 self)
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Abstract—We generalize the notions of centroids (and barycenters) to the broad class of informationtheoretic distortion measures called Bregman divergences. Bregman divergences form a rich and versatile family of distances that unifies quadratic Euclidean distances with various wellknown statistical entropic measures. Since besides the squared Euclidean distance, Bregman divergences are asymmetric, we consider the leftsided and rightsided centroids and the symmetrized centroids as minimizers of average Bregman distortions. We prove that all three centroids are unique and give closedform solutions for the sided centroids that are generalized means. Furthermore, we design a provably fast and efficient arbitrary close approximation algorithm for the symmetrized centroid based on its exact geometric characterization. The geometric approximation algorithm requires only to walk on a geodesic linking the two left/right sided centroids. We report on our implementation for computing entropic centers of image histogram clusters and entropic centers of multivariate normal distributions that are useful operations for processing multimedia information and retrieval. These experiments illustrate that our generic methods compare favorably with former limited adhoc methods. Index Terms—Centroid, KullbackLeibler divergence, Bregman divergence, Bregman power divergence, BurbeaRao divergence,
Hyperbolic Voronoi diagrams made easy
, 2009
"... We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein’s nonconformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. Therefore our method simply consists in computing ..."
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Cited by 20 (7 self)
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We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein’s nonconformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. Therefore our method simply consists in computing an equivalent clipped power diagram followed by a mapping transformation depending on the selected representation of the hyperbolic space (e.g., Poincaré conformal disk or upperplane representations). We discuss on extensions of this approach to weighted and korder diagrams, and describe their dual triangulations. Finally, we consider two useful primitives on the hyperbolic Voronoi diagrams for designing tailored user interfaces of an image catalog browsing application in the hyperbolic disk: (1) finding nearest neighbors, and (2) computing smallest enclosing balls.
Shape Retrieval Using Hierarchical Total Bregman Soft Clustering
"... In this paper, we consider the family of total Bregman divergences (tBDs) as an efficient and robust “distance” measure to quantify the dissimilarity between shapes. We use the tBD based ℓ1norm center as the representative of a set of shapes, and call it the tcenter. First, we briefly present and ..."
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Cited by 14 (4 self)
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In this paper, we consider the family of total Bregman divergences (tBDs) as an efficient and robust “distance” measure to quantify the dissimilarity between shapes. We use the tBD based ℓ1norm center as the representative of a set of shapes, and call it the tcenter. First, we briefly present and analyze the properties of the tBDs and tcenters following our previous work in [1]. Then, we prove that for any tBD, there exists a distribution which belongs to the lifted exponential family of statistical distributions. Further, we show that finding the maximum a posteriori estimate of the parameters of the lifted exponential family distribution is equivalent to minimizing the tBD to find the tcenters. This leads to a new clustering technique namely, the total Bregman soft clustering algorithm. We evaluate the tBD, tcenter and the soft clustering algorithm on shape retrieval applications. Our shape retrieval framework is composed of three steps: (1) extraction of the shape boundary points (2) affine alignment of the shapes and use of a Gaussian mixture model (GMM) [2], [3], [4] to represent the aligned boundaries, and (3) comparison of the GMMs using tBD to find the best matches given a query shape. To further speed up the shape retrieval algorithm, we perform hierarchical clustering of the shapes using our total Bregman soft clustering algorithm. This enables us to compare the query with a small subset of shapes which are chosen to be the cluster tcenters. We evaluate our method on various public domain 2D and 3D databases, and demonstrate comparable or better results than stateoftheart retrieval techniques.
APPROXIMATING SMALLEST ENCLOSING BALLS WITH APPLICATIONS TO MACHINE LEARNING
, 2007
"... In this paper, we first survey prior work for computing exactly or approximately the smallest enclosing balls of point or ball sets in Euclidean spaces. We classify previous work into three categories: (1) purely combinatorial, (2) purely numerical, and (3)recent mixed hybrid algorithms based on cor ..."
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Cited by 10 (1 self)
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In this paper, we first survey prior work for computing exactly or approximately the smallest enclosing balls of point or ball sets in Euclidean spaces. We classify previous work into three categories: (1) purely combinatorial, (2) purely numerical, and (3)recent mixed hybrid algorithms based on coresets. We then describe two novel tailored algorithms for computing arbitrary close approximations of the smallest enclosing Euclidean ball of balls. These deterministic heuristics are based on solving relaxed decision problems using a primaldual method. The primaldual method is interpreted geometrically as solving for a minimum covering set, or dually as seeking for a minimum piercing set. Finally, we present some applications in machine learning of the exact and approximate smallest enclosing ball procedure, and discuss about its extension to nonEuclidean informationtheoretic spaces.
Total Bregman Divergence and its Applications to Shape Retrieval ∗
"... Shape database search is ubiquitous in the world of biometric systems, CAD systems etc. Shape data in these domains is experiencing an explosive growth and usually requires search of whole shape databases to retrieve the best matches with accuracy and efficiency for a variety of tasks. In this paper ..."
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Cited by 9 (4 self)
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Shape database search is ubiquitous in the world of biometric systems, CAD systems etc. Shape data in these domains is experiencing an explosive growth and usually requires search of whole shape databases to retrieve the best matches with accuracy and efficiency for a variety of tasks. In this paper, we present a novel divergence measure between any two given points in Rn or two distribution functions. This divergence measures the orthogonal distance between the tangent to the convex function (used in the definition of the divergence) at one of its input arguments and its second argument. This is in contrast to the ordinate distance taken in the usual definition of the Bregman class of divergences [4]. We use this orthogonal distance to redefine the Bregman class of divergences and develop a new theory
Sparse multiscale patches for image processing
 In ETVC, volume 5416/2009 of LNCS
, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 8 (5 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Approximate Bregman near neighbors in sublinear time: Beyond the triangle inequality
 CoRR
"... Bregman divergences are important distance measures that are used extensively in datadriven applications such as computer vision, text mining, and speech processing, and are a key focus of interest in machine learning. Answering nearest neighbor (NN) queries under these measures is very important ..."
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Cited by 3 (1 self)
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Bregman divergences are important distance measures that are used extensively in datadriven applications such as computer vision, text mining, and speech processing, and are a key focus of interest in machine learning. Answering nearest neighbor (NN) queries under these measures is very important in these applications and has been the subject of extensive study, but is problematic because these distance measures lack metric properties like symmetry and the triangle inequality. In this paper, we present the first provably approximate nearestneighbor (ANN) algorithms. These process queries in O(logn) time for Bregman divergences in fixed dimensional spaces. We also obtain poly logn bounds for a more abstract class of distance measures (containing Bregman divergences) which satisfy certain structural properties. Both of these bounds apply to both the regular asymmetric Bregman divergences as well as their symmetrized versions. To do so, we develop two geometric properties vital to our analysis: a reverse triangle inequality (RTI) and a relaxed triangle inequality called µdefectiveness where µ is a domaindependent parameter. Bregman divergences satisfy the RTI but not µdefectiveness. However, we show that the square root of a Bregman divergence does satisfy µdefectiveness. This allows us to then utilize both properties in an efficient search data structure that follows the general twostage paradigm of a ringtree decomposition followed by a quad tree search used in previous nearneighbor algorithms for Euclidean space and spaces of bounded doubling dimension. Our first algorithm resolves a query for a ddimensional (1+ ε)ANN in O ( lognε)
Smallest Enclosing Ball for a Point Set with Strictly Convex Level Sets
, 2007
"... Let the center point be the point that minimizes the maximum distance from a point of a given point set to the center point. Finding this center point is referred to as the smallest enclosing ball problem. In case of points with Euclidean distance functions, the smallest enclosing ball is actually t ..."
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Cited by 1 (0 self)
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Let the center point be the point that minimizes the maximum distance from a point of a given point set to the center point. Finding this center point is referred to as the smallest enclosing ball problem. In case of points with Euclidean distance functions, the smallest enclosing ball is actually the center of a geometrical ball. We consider point sets with points that have distance functions with strictly convex level sets. For such point sets, we show that the smallest enclosing ball exists, is unique, and can be computed using an algorithm for solving LPtype problems as it was introduced by Sharir and Welzl in [34]. We provide a list of distance functions, show that they have strictly convex level sets, and hint at the implementations of the basic operations used by the LPtype algorithm. In a last part, we analyze approximative solutions of the smallest enclosing ball problem and conclude that there are no ɛcore sets for some of the considered
Chapter 1 Chebyshev Sets, Klee Sets, and Chebyshev Centers with respect to Bregman Distances: Recent Results and Open Problems
, 2010
"... Abstract In Euclidean spaces, the geometric notions of nearestpoints map, farthestpoints map, Chebyshev set, Klee set, and Chebyshev center are well known and well understood. Since early works going back to the 1930s, tremendous theoretical progress has been made, mostly by extending classical re ..."
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Abstract In Euclidean spaces, the geometric notions of nearestpoints map, farthestpoints map, Chebyshev set, Klee set, and Chebyshev center are well known and well understood. Since early works going back to the 1930s, tremendous theoretical progress has been made, mostly by extending classical results from Euclidean space to Banach space settings. In all these results, the distance between points is induced by some underlying norm. Recently, these notions have been revisited from a different viewpoint in which the discrepancy between points is measured by Bregman distances induced by Legendre functions. The associated framework covers the well known KullbackLeibler divergence and the ItakuraSaito distance. In this survey, we review known results and we present new results on Klee sets and Chebyshev centers with respect to Bregman distances. Examples are provided and connections to recent work on Chebyshev functions are made. We also identify several intriguing open problems.