We define a class of L-convex-concave subsets of RP n , where L is a projective sub-space of dimension l in RP n. These are sets whose sections by any (l+1)-dimensional space L ′ containing L are convex and concavely depend on L ′. We introduce an L-duality for these sets, and prove that the L-dual to an L-convex-concave set is an L *-convex-concave subset of (RP n) *. We discuss a version of A...

We examine models for unsupervised learning with concave log-likelihood functions. We begin with the most well-known example, IBM Model 1 for word alignment (Brown et al., 1993), and study its properties, discussing why other models for unsupervised learning are so seldom concave. We then present concave models for dependency grammar induction and validate them experimentally. Despite their sim...

In this paper, we introduce a more general approach to the fuzzification of fuzzy concavity. More specifically, degree L , M -fuzzy concavity is introduced and characterized as generalization id="M2"> -concave structure id="M3"> conca...

In this paper we consider quasi-concave set functions defined on antimatroids. There are many equivalent axiomatizations of antimatroids, that may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. An algorthmic characterization of antimatroids, that considers them as set systems, was given in [4]. This characterization is based on the i...

This is a small observation concerning scale mixtures and their log-concavity. A function f(x) ≥ 0, x ∈ Rn is called log-concave if f (λx + (1− λ)y) ≥ f(x)f(y) (1) for all x,y ∈ Rn, λ ∈ [0, 1]. Log-concavity is important in applied Bayesian Statistics, since a distribution with a log-concave density is easy to treat with many different approximate inference techniques. For example, log-concavit...

The fuzzy integrals are a kind of fuzzy measures acting on fuzzy sets. They can be viewed as an average membershipvalue of fuzzy sets. The value of the fuzzy integral in a decision making environment where uncertainty is presenthas been well established. Most of the integral inequalities studied in the fuzzy integration context normally considerconditions such as monotonicity or comonotonicity....