We present an extension of Fenchel’s duality theorem to nearly convexity, giving weaker conditions under which it takes place. Instead of minimizing the difference between a convex and a concave function, we minimize the subtraction of a nearly concave function from a nearly convex one. The assertion in the special case of Fenchel’s duality theorem that consists in minimizing the difference bet...

In this paper, we prove that every multivariate polynomial with even degree can be decomposed into a sum of convex and concave polynomials. Motivated by this property, we exploit the concave-convex procedure to perform inference on continuous Markov random fields with polynomial potentials. In particular, we show that the concave-convex decomposition of polynomials can be expressed as a sum-of-...

Concave compositions were recently introduced by Andrews[3] in the study of orthogonal polynomials, see also Andrews [4]. A concave composition of even length 2m, is a sum of the form ∑ ai + ∑ bi such that a1 > a2 > · · · > am = bm < bm−1 < · · · < b1, where am ≥ 0, and all ai and bi are integers. Let CE(n) denote the set of concave compositions of even length that sum to n, and ce(n) be the ca...

Mixed f -divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov-Fenchel type inequalities and an affine isoperimetric inequality for the vector form of the Kull...

We discuss the relationship between matroid rank functions and a concept of discrete concavity called M-concavity. It is known that a matroid rank function and its weighted version called a weighted rank function are M-concave functions, while the (weighted) sum of matroid rank functions is not M-concave in general. We present a sufficient condition for a weighted sum of matroid rank functions ...

A general framework based on majorization, Schur-concavity, and concavity is given that facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed diversity measures useful for best basis selection. Admissible sparsity measures are given by the Schur-concave functions, which are the class of functions consistent with the partial ordering on vecto...

A way of making Bayesian inference for concave distribution functions is introduced. This is done by uniquely transforming a mixture of Dirichlet processes on the space of distribution functions to the space of concave distribution functions. The approach also gives a way of making Bayesian analysis of mul-tiplicatively censored data. We give a method for sampling from the posterior distributio...

Disclinations, concave and convex, are the canonical point defects of twodimensional planar patterns in systems with translational and rotational symmetries. From these, all other point defects (vortices, dislocations, targets, saddles and handles) can be built. Moreover, handles, coupled concave–convex disclination pairs arise as instabilities, symmetry breaking events. The purpose of this art...

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The article presents conditions under which the probability of a linear combination random vectors falling into polyhedral cone is Schur-concave function coefficients combination. It required that contains point 0, its edges are parallel to coordinate axes, and distribution density logarithmically concave sign-invariant function.