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The Dual Voronoi Diagrams with Respect to Representational Bregman Divergences

Frank Nielsen, Richard Nock
2009 2009 Sixth International Symposium on Voronoi Diagrams  
with respect to the dual coordinate systems x/x * .  ...  1 −x β(β+1) U * β (x) = x β+1 −x β(β+1) k * β (x) = Generalized Bregman Voronoi diagrams as lower envelopes Voronoi diagrams obtained as minimization diagrams of functions: min i∈{1,...  ...  Voronoi diagrams of representable Bregman divergences Theorem. The Voronoi diagram of n d-dimensional points with respect to a representational Bregman divergence has complexity O(n ⌈ d 2 ⌉ ).  ... 
doi:10.1109/isvd.2009.15 dblp:conf/isvd/NielsenN09 fatcat:k63x2d2zh5eg7bdp6fpmo7euz4

Quantum Voronoi diagrams and Holevo channel capacity for 1-qubit quantum states

Frank Nielsen, Richard Nock
2008 2008 IEEE International Symposium on Information Theory  
Bloch sphere of pure quantum states coincides with the ordinary Euclidean Voronoi diagram, bypassing the fact that the quantum divergence is not defined there.  ...  This implies that these dual diagrams can be derived from power diagrams of balls in the Laguerre geometry, and allows one to prove by equivalence that the von Neumann quantum Voronoi diagram on the degenerated  ...  [7] , [8] investigated the Voronoi diagram with respect to the quantum information divergence and carried out calculations in the limit case of Q being a pure state.  ... 
doi:10.1109/isit.2008.4594955 dblp:conf/isit/NielsenN08 fatcat:5pe5nltwhfhdraktmfkqoen4t4

Bregman Voronoi Diagrams

Jean-Daniel Boissonnat, Frank Nielsen, Richard Nock
2010 Discrete & Computational Geometry  
We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams  ...  Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions.  ...  We thank the anonymous reviewers for their constructive comments.  ... 
doi:10.1007/s00454-010-9256-1 fatcat:lik3travqzdfhbm2md3p2t5mfi

Bregman Voronoi Diagrams: Properties, Algorithms and Applications [article]

Frank Nielsen, Jean-Daniel Boissonnat, Richard Nock
2007 arXiv   pre-print
We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams  ...  Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions.  ...  The work by the second author has been partially supported by the project GeoTopAl (1555) of the Agence Nationale de la Recherche (ANR).  ... 
arXiv:0709.2196v1 fatcat:7juz25kny5fi3g45qkag3c4xze

On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds [article]

Frank Nielsen
2020 arXiv   pre-print
The dual Voronoi diagrams with respect to the dual forward/reverse flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations.  ...  The primal Bregman-Tsallis Voronoi diagram corresponds to the hyperbolic Voronoi diagram and the dual Bregman-Tsallis Voronoi diagram coincides with the ordinary Euclidean Voronoi diagram.  ...  The dual Voronoi diagrams with respect to the dual flat divergences can be built from the corresponding dual Bregman-Tsallis divergences with the primal Voronoi diagram coinciding with the hyperbolic Voronoi  ... 
arXiv:2006.07020v2 fatcat:cz7vqlr5tfgynfn4njgs5qhwwu

On Voronoi Diagrams on the Information-Geometric Cauchy Manifolds

Frank Nielsen
2020 Entropy  
The dual Voronoi diagrams with respect to the dual flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations.  ...  The primal Bregman Voronoi diagram is the Euclidean Voronoi diagram and the dual Bregman Voronoi diagram coincides with the Cauchy hyperbolic Voronoi diagram.  ...  The dual Voronoi diagrams with respect to the dual flat divergences can be built from the corresponding dual Bregman-Tsallis divergences with the primal Voronoi diagram coinciding with the hyperbolic Voronoi  ... 
doi:10.3390/e22070713 pmid:33286486 pmcid:PMC7517249 fatcat:7afynnec6jedfenemlydhxhplm

Pattern Learning and Recognition on Statistical Manifolds: An Information-Geometric Review [chapter]

Frank Nielsen
2013 Lecture Notes in Computer Science  
We explain the construction of the two prominent families of statistical manifolds: The Rao Riemannian manifolds with geodesic metric distances, and the Amari-Chentsov manifolds with dual asymmetric non-metric  ...  divergences.  ...  For distributions belonging to the same exponential families, those statistical Voronoi diagrams amount to perform Bregman Voronoi diagrams on the distribution parameters (using either the natural θ-coordinates  ... 
doi:10.1007/978-3-642-39140-8_1 fatcat:g5cxsyc6q5gpdnme6ijoe7omuq

On the Embeddability of Delaunay Triangulations in Anisotropic, Normed, and Bregman Spaces [article]

Guillermo D. Canas, Steven J. Gortler
2016 arXiv   pre-print
We then prove that the straight-edge dual of an orphan-free Voronoi diagram (with sites as the first argument of the divergence) is always an embedded triangulation.  ...  While Bregman diagrams of the first kind are simply affine diagrams, and their duals (weighted Delaunay triangulations) are always embedded, we show that duals of orphan-free Bregman diagrams of the second  ...  The recent work of [3] discusses Voronoi diagrams and their duals with respect to Bregman divergences.  ... 
arXiv:1512.03589v2 fatcat:e4yie73h55c7jjlvrwap3vacfu

Fast nearest neighbor retrieval for bregman divergences

Lawrence Cayton
2008 Proceedings of the 25th international conference on Machine learning - ICML '08  
The family of bregman divergences includes many popular dissimilarity measures including KL-divergence (relative entropy), Mahalanobis distance, and Itakura-Saito divergence.  ...  The data structure introduced in this work shares the same basic structure as the popular metric ball tree, but employs convexity properties of bregman divergences in place of the triangle inequality.  ...  Support was provided by the NSF under grants IIS-0347646 and IIS-0713540.  ... 
doi:10.1145/1390156.1390171 dblp:conf/icml/Cayton08 fatcat:cjny6hequzhvfbabk224oxemcq

Voronoi Diagrams for Quantum States and Its Application to a Numerical Estimation of a Quantum Channel Capacity [article]

Kimikazu Kato
2008 arXiv   pre-print
Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states.  ...  With those tools in computational geometry, we analyze the adjacency structure of a point set in the quantum state space.  ...  diagram with respect to the Euclidean distance 3. the Voronoi diagram with respect to the divergences, i.e.  ... 
arXiv:0803.3109v2 fatcat:k3mtauzkt5gyde3hbim5scscwi

An elementary introduction to information geometry [article]

Frank Nielsen
2020 arXiv   pre-print
The exposition is self-contained by concisely introducing the necessary concepts of differential geometry, but proofs are omitted for brevity.  ...  In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information  ...  On the dually flat exponential family manifold, it corresponds to check the exponential arcs between natural neighbors (sharing Voronoi subfaces) of a Bregman Voronoi diagram [20].  ... 
arXiv:1808.08271v2 fatcat:q7cum6qk7ffbhpbmmyynvdeeya

An Elementary Introduction to Information Geometry

Frank Nielsen
2020 Entropy  
The exposition is self-contained by concisely introducing the necessary concepts of differential geometry. Proofs are omitted for brevity.  ...  In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information  ...  The Bregman spheres are considered for building Bregman Voronoi diagrams in [40, 42] .  ... 
doi:10.3390/e22101100 pmid:33286868 pmcid:PMC7650632 fatcat:ye2h3cjx35e6xcbrhiic2hmuea

Approximate Bregman near neighbors in sublinear time: Beyond the triangle inequality [article]

Amirali Abdullah, John Moeller, Suresh Venkatasubramanian
2013 arXiv   pre-print
The second algorithm processes queries in O(log n) time using O(n) space and exploits structural constants associated specifically with Bregman divergences.  ...  In this paper we present the first provable approximate nearest-neighbor (ANN) algorithms for Bregman divergences.  ...  We note in passing that such problems do not occur for the exact nearest neighbor problem in constant dimension: this problem reduces to point location in a Voronoi diagram, and Bregman Voronoi diagrams  ... 
arXiv:1108.0835v7 fatcat:jkqt2gw6hbhsdfnapk2rqxelci

Abstracts of the LIX Fall Colloquium 2008: Emerging Trends in Visual Computing [chapter]

Frank Nielsen
2009 Lecture Notes in Computer Science  
It is also of interest to perform such structure discovery across multiple data sets distributed in a network, without actually ever bring them all to the same host.  ...  In most instances, the original scanner data is just a set, but a very large set, of points sampled from the surface of the object.  ...  Namely, we introduce the generic classes of Bregman, Csiszar and Burbea-Rao parametric divergences and explain their relationships and properties with respect to algorithmic design.  ... 
doi:10.1007/978-3-642-00826-9_1 fatcat:7ppk5awzqrhexlrcvbon24qr7m

Hartigan's Method for $$k$$ k -MLE: Mixture Modeling with Wishart Distributions and Its Application to Motion Retrieval [chapter]

Christophe Saint-Jean, Frank Nielsen
2014 Signals and Communication Technology  
to the MLE.  ...  Finally, we propose to use the Cauchy-Schwartz statistical divergence as a dissimilarity measure between two Wishart mixture models and sketch a general methodology for building a motion retrieval system  ...  weight − log w j (weighted Bregman Voronoi diagram [29] ).  ... 
doi:10.1007/978-3-319-05317-2_11 fatcat:xva2mvgxcrgzhcpn5nuxrbbxki
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