Bregman Voronoi Diagrams

Boissonnat, Jean‐Daniel; Nielsen, Frank; Nock, Richard

doi:10.1007/s00454-010-9256-1

Cited by 92 publications

(127 citation statements)

References 31 publications

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“…The basic Beta-divergence was introduced by Basu et al [67] and Minami and Eguchi [15] and many researchers investigated their applications including [8,13,[30][31][32][33][34]37,37,[68][69][70][71][72], and references therein. The main motivation to investigate the beta divergence, at least from the practical point of view, is to develop highly robust in respect to outliers learning algorithms for clustering, feature extraction, classification and blind source separation.…”

Section: Family Of Beta-divergencesmentioning

confidence: 99%

“…For example, in sound processing, the speech power spectra can be modeled by exponential family densities of the form, whose for β = 0 the Beta-divergence is no less than the Itakura-Saito distance (called also Itakura-Saito divergence or Itakura-Saito distortion measure or Burg cross entropy) [12,13,30,[76][77][78][79]. In fact, the Beta-divergence has to be defined in limiting case for β → 0 as the Itakura-Saito distance:…”

Section: Family Of Beta-divergencesmentioning

confidence: 99%

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Families of Alpha- Beta- and Gamma- Divergences: Flexible and Robust Measures of Similarities

Cichocki

Амари

2010

Entropy

381

313

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In this paper, we extend and overview wide families of Alpha-, Beta-and Gamma-divergences and discuss their fundamental properties. In literature usually only one single asymmetric (Alpha, Beta or Gamma) divergence is considered. We show in this paper that there exist families of such divergences with the same consistent properties. Moreover, we establish links and correspondences among these divergences by applying suitable nonlinear transformations. For example, we can generate the Beta-divergences directly from Alpha-divergences and vice versa. Furthermore, we show that a new wide class of Gamma-divergences can be generated not only from the family of Beta-divergences but also from a family of Alpha-divergences. The paper bridges these divergences and shows also their links to Tsallis and Rényi entropies. Most of these divergences have a natural information theoretic interpretation.

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Section: Family Of Beta-divergencesmentioning

confidence: 99%

Section: Family Of Beta-divergencesmentioning

confidence: 99%

Families of Alpha- Beta- and Gamma- Divergences: Flexible and Robust Measures of Similarities

Cichocki

Амари

2010

Entropy

381

313

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“…A similar situation is known for certain standard non-Euclidean geometries, such as Laguerre geometry [Aur87], spaces equipped with a Bregman divergence [BNN10], or Riemannian manifolds of constant sectional curvature 1 .…”

Section: Delaunay Complex and Delaunay Triangulationmentioning

confidence: 80%

An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Boissonnat

Dyer²,

Ghosh

et al. 2017

Discrete Comput Geom

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Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space R m triangulates the convex hull of P , provided that P satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on P are required. A natural one is to assume that P is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone is insufficient to ensure that the Delaunay complex triangulates a manifold of dimension greater than 2. Delaunay complex and Delaunay triangulationLet (M, d M ) be a metric space, and let P be a finite set of points of M. An empty ball is an open ball in the metric d M that contains no point from P . We say that an empty ball B is maximal if no other empty ball with the same centre properly contains B. A Delaunay ball is a maximal empty ball.A simplex σ is a Delaunay simplex if there exists some Delaunay ball B that circumscribes σ, i.e., such that the vertices of σ belong to ∂B ∩ P . The Delaunay complex is the set of Delaunay simplices, and is denoted Del M (P ). It is an abstract simplicial complex and so defines a topological space, |Del M (P )|, called its carrier . We say that DelThe Voronoi cell associated with p ∈ P is given byMore generally, a Voronoi face is the intersection of a set of Voronoi cells: given σ = {p 0 , . . . , p k } ⊂ P , we define the associated Voronoi face asIt follows that σ is a Delaunay simplex if and only if V M (σ) = ∅. In this case, every point in V M (σ) is the centre of a Delaunay ball for σ. Thus every Voronoi face corresponds to a Delaunay

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“…This bisector is a hyperplane in the η = ∇F (θ) coordinate system [16] (but a hypersurface in the θ-coordinate system), hence its name m-bisector Bi m (P 1 , P 2 ). It follows that P * = G e (P 1 , P 2 ) ∩ Bi m (P 1 , P 2 ).…”

Section: A Geometric Characterization Of the Chernoff Distributionmentioning

confidence: 99%

An Information-Geometric Characterization of Chernoff Information

Nielsen

2013

IEEE Signal Process. Lett.

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The Chernoff information was originally introduced for bounding the probability of error of the Bayesian decision rule in binary hypothesis testing. Nowadays, it is often used as a notion of symmetric distance in statistical signal processing or as a way to define a middle distribution in information fusion.Computing the Chernoff information requires to solve an optimization problem that is numerically approximated in practice. We consider the Chernoff distance for distributions belonging to the same exponential family including the Gaussian and multinomial families. By considering the geometry of the underlying statistical manifold, we define exactly the solution of the optimization problem as the unique intersection of a geodesic with a dual hyperplane. Furthermore, we prove analytically that the Chernoff distance amounts to calculate an equivalent but simpler Bregman divergence defined on the distribution parameters. It follows a closed-form formula for the singly-parametric distributions, or an efficient geodesic bisection search for multi-parametric distributions. Finally, based on this informationgeometric characterization, we propose three novel information-theoretic symmetric distances and middle distributions, from which two of them admit always closed-form expressions.

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Bregman Voronoi Diagrams

Cited by 92 publications

References 31 publications

Families of Alpha- Beta- and Gamma- Divergences: Flexible and Robust Measures of Similarities

Families of Alpha- Beta- and Gamma- Divergences: Flexible and Robust Measures of Similarities

An Obstruction to Delaunay Triangulations in Riemannian Manifolds

An Information-Geometric Characterization of Chernoff Information

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