Carathéodory's Theorem and H-Convexity

Boltyanski, Vladimir G.; Martini, Horst

doi:10.1006/jcta.2000.3080

Cited by 8 publications

(4 citation statements)

References 14 publications

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“…As we have seen y is a barycenter of (f, g)-achievable points that are 2achievable. According to the Carathéodory's theorem [12] any barycentric decomposition in two dimensions may be obtained as a convex combination of at most three points y i , i = 1, 2, 3. as illustrated in Figure 3. Assume that all three points have positive weight.…”

Section: Joint Range Of F -Divergencesmentioning

confidence: 99%

On Pairs of $f$-Divergences and Their Joint Range

Harremoës

Vajda

2011

IEEE Trans. Inform. Theory

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Section: Joint Range Of F -Divergencesmentioning

confidence: 99%

On Pairs of $f$-Divergences and Their Joint Range

Harremoës

Vajda

2011

IEEE Trans. Inform. Theory

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“…(ii) As a next step, it would be natural to extend the present investigations to generalized convexity notions, like for example H-convexity (see, e.g., [1]…”

Section: Carathéodory's and Radon's Theoremmentioning

confidence: 99%

Helly's Theorem and Its Equivalences via Convex Analysis

Robinson

2014

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Helly's theorem is an important result from Convex Geometry. It gives sufficient conditions for a family of convex sets to have a nonempty intersection. A large variety of proofs as well as applications are known. Helly's theorem also has close connections to two other well-known theorems from Convex Geometry: Radon's theorem and Carathéodory's theorem. In this project we study Helly's theorem and its relations to Radon's theorem and Carathéodory's theorem by using tools of Convex Analysis and Optimization. More precisely, we will give a novel proof of Helly's theorem, and in addition we show in a complete way that these three famous theorems are equivalent in the sense that using one of them allows us to derive the others. Contents Ch. 1. Elements of Convex Analysis and Optimization 3 Ch.

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“…As we have seen y is a barycenter of (f, g)-divergence pairs achievable in R 2 . According to the Carathéodory's theorem [12] any barycentric decomposition in two dimensions may be obtained as a convex combination of at most three points y i , i = 1, 2, 3. as illustrated in Figure 3. Assume that all three points have positive weight.…”

Section: Theoremmentioning

confidence: 99%

Joint range of f-divergences

Harremoës

Vajda

2010

2010 IEEE International Symposium on Information Theory

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I. DIVERGENCES AND DIVERGENCE STATISTICSMany of the divergence measures used in statistics are of the f -divergence type introduced independently by I. Csiszár [1], T. Morimoto [2], and Ali and Silvey [3]. Such divergence measures have been studied in great detail in [4]. Often one is interested inequalities for one f -divergence in terms of another f -divergence. Such inequalities are for instance needed in order to calculate the relative efficiency of two fdivergences when used for testing goodness of fit but there are many other applications. In this paper we shall study the more general problem of determining the joint range of any pair of f -divergences. The results are useful in determining general conditions under which information divergence is a more efficient statistic for testing goodness of fit than another f -divergence, but will not be discussed in this short paper.Lett . Assume that P and Q are absolutely continuous with respect to a measure µ, and that p = dP dµ and q = dQ dµ . For arbitrary distributions P and Q the f -divergence D f (P, Q) ≥ 0 is defined by the formula1) (for details about the definition (1) and properties of the fdivergences, see [5], [4] or [6]). With this definitionExample 1: The function f (t) = |t − 1| defines the L 1 -distancewhich plays an important role in information theory and mathematical statistics (cf.[7] or [8]). In (1) is often taken the convex function f which is one of the power functions φ α of order α ∈ R given in the domain t > 0 by the formulaα(α − 1) = 0 (3) D(P Q) V (P, Q) 2 3 1 0 1 2 Fig. 1. The joint range of total variation V and information D as determined in [8]. It was also proved that any point in the range and by the corresponding limits φ 0 (t) = − ln t + t − 1 and φ 1 (t) = t ln t − t + 1. (4)The φ-divergencesbased on (3) and (4) are usually referred to as power divergences of orders α. For details about the properties of power divergences, see [5] or [6]. Next we mention the best known members of the family of statistics (5), with a reference to the skew symmetry D α (P, Q) = D 1−α (Q, P ) of the power divergences (5).Example 2: The χ 2 -divergence or quadratic divergence D 2 (P, Q) = D −1 (Q, P ) = 1 2 k j=1 (p j − q j ) 2 q j

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Carathéodory's Theorem and H-Convexity

Cited by 8 publications

References 14 publications

On Pairs of $f$-Divergences and Their Joint Range

On Pairs of $f$-Divergences and Their Joint Range

Helly's Theorem and Its Equivalences via Convex Analysis

Joint range of f-divergences

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