We introduce the Concave-Convex procedure (CCCP) which constructs discrete time iterative dynamical systems which are guaranteed to monotonically decrease global optimization/energy functions. It can be applied to (almost) any optimization problem and many existing algorithms can be interpreted in terms of CCCP. In particular, we prove relationships to some applications of Legendre transform te...

The Minimum Cost Network Flow Problem (MCNFP) includes a wide range of combinatorial optimization problems. Many applications exist for MCNFPs for instance supply chains, logistics, production planning, communications and transportations. Concave costs are, in many applications, more realistic than linear ones because of the association of prices with economies of scale. When concave costs are ...

Finite mixtures of parametric distributions are often used to model data of which it is known or suspected that there are subpopulations. Instead of a parametric model, a penalized likelihood smoothing algorithm is developed. The penalty is chosen to favor a log-concave result. The standard EM algorithm (“split and fit”) can be used. Theoretical results and applications are presented. © 2006 El...

An integer partition λ ` n corresponds, via its Ferrers diagram, to an artinian monomial ideal I ⊂ C[x, y] with dimC C[x, y]/I = n. If λ corresponds to an integrally closed ideal we call it concave . We study generating functions for the number of concave partitions, unrestricted or with at most r parts. 1. concave partitions By an integer partition λ = (λ1, λ2, λ3, . . . ) we mean a weakly dec...

A convex triangular grid is a planar digraph G embedded in the plane so that each bounded face is an equilateral triangle with three edges and their union R forms a convex polygon. A function h : E(G) → R is called a concave cocirculation if h(e) = g(v) − g(u) for each edge e = (u, v), where g is a concave function on R which is affinely linear within each bounded face of G. Knutson and Tao obt...

We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on Rd, for all d ≥ 1. Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of d > 3. In more detail, we give an estimator that, for any d ≥ 1 and ǫ > 0, draws Õd ( (1/ǫ...

We prove that the reciprocal of Fisher information of a logconcave probability density X in Rn is concave in t with respect to the addition of a Gaussian noise Zt = N(0, tIn). As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density X in Rn is nonnegative in t with respect to the addition of a Gaussian noise Zt. For log-concave de...

Log-concave distributions are an attractive choice for modeling and inference, for several reasons: The class of log-concave distributions contains most of the commonly used parametric distributions and thus is a rich and flexible nonparametric class of distributions. Further, the MLE exists and can be computed with readily available algorithms. Thus, no tuning parameter, such as a bandwidth, i...

Copyright q 2010 Y.-M. Chu and X.-M. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove that G x, y | ∫x y f t dt| is multiplicatively concave on a, b × a, b if f : a, b ⊂ 0,∞ → 0,∞ is continuous and multiplicatively ...