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The Dual Voronoi Diagrams with Respect to Representational Bregman Divergences

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 ⌉ ). ...

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

2008
*
2008 IEEE International Symposium on Information Theory
*

Bloch sphere of pure quantum states coincides

doi:10.1109/isit.2008.4594955
dblp:conf/isit/NielsenN08
fatcat:5pe5nltwhfhdraktmfkqoen4t4
*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. ...##
###
Bregman Voronoi Diagrams

2010
*
Discrete & Computational Geometry
*

We also introduce extensions of these

doi:10.1007/s00454-010-9256-1
fatcat:lik3travqzdfhbm2md3p2t5mfi
*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. ...##
###
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
[article]

2007
*
arXiv
*
pre-print

We also introduce extensions of these

arXiv:0709.2196v1
fatcat:7juz25kny5fi3g45qkag3c4xze
*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). ...##
###
On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds
[article]

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*...

##
###
On Voronoi Diagrams on the Information-Geometric Cauchy Manifolds

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*...

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

2013
*
Lecture Notes in Computer Science
*

We explain

doi:10.1007/978-3-642-39140-8_1
fatcat:g5cxsyc6q5gpdnme6ijoe7omuq
*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 ...##
###
On the Embeddability of Delaunay Triangulations in Anisotropic, Normed, and Bregman Spaces
[article]

2016
*
arXiv
*
pre-print

We then prove that

arXiv:1512.03589v2
fatcat:e4yie73h55c7jjlvrwap3vacfu
*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*. ...##
###
Fast nearest neighbor retrieval for bregman divergences

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. ...

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

2008
*
arXiv
*
pre-print

Especially we introduce

arXiv:0803.3109v2
fatcat:k3mtauzkt5gyde3hbim5scscwi
*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. ...##
###
An elementary introduction to information geometry
[article]

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]. ...

##
###
An Elementary Introduction to Information Geometry

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] . ...

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

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*...

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

2009
*
Lecture Notes in Computer Science
*

It is also of interest

doi:10.1007/978-3-642-00826-9_1
fatcat:7ppk5awzqrhexlrcvbon24qr7m
*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. ...##
###
Hartigan's Method for $$k$$ k -MLE: Mixture Modeling with Wishart Distributions and Its Application to Motion Retrieval
[chapter]

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] ). ...

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