We study the convexity of mutual information along the evolution of the heat equation. We prove that if the initial distribution is log-concave, then mutual information is always a convex function of time. We also prove that if the initial distribution is either bounded, or has finite fourth moment and Fisher information, then mutual information is eventually convex, i.e., convex for all large ...

Given a set S of n distinct points { (xi'Yi) 1 0 5 i < n] 9 the convex hull problem is to determine the vertices of the convex hull H(S) . All the known algorithms for solving this problem have a worst-case running time of cn log n or higher, and employ only quadratic tests, i.e., tests of ‘the form f(Xo’YO ⌧19 Yl> l l l ☺ ⌧nBl’ Y, 1> : 0 with f being any polynomial of degree not exceeding 2 . ...

In the paper, after reviewing the history, background, origin, and applications of the functions b t −a t t and e −αt −e −βt 1−e−t , we establish sufficient and necessary conditions such that the special function e αt −e βt eλt−eμt are monotonic, logarithmic convex, logarithmic concave, 3-log-convex and 3-log-concave on R, where α, β, λ and μ are real numbers satisfying (α, β) 6= (λ, μ), (α, β)...

We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this paper, we pose it as a combinatorial optimization problem, which is then relaxed to a convex optimization problem that involves searching over the forest and hype...

We prove the connexity of the boundary of each 4-dimensional cell of the 3D visibility complex of disjoint convex polyhedra. Using this property, we outline a simple algorithm for the construction of the visibility complex of k disjoint, convex polyhedra in time O(nk log n) where n is the total complexity of the polyhedra.

It is known that the bandwidth problem is NP-complete for chordal bipartite graphs, while the problem can be solved in polynomial time for bipartite permutation graphs, which is a subclass of chordal bipartite graphs. This paper shows that the problem is NP-complete even for convex bipartite graphs, a subclass of chordal bipartite graphs and a superclass of bipartite permutation graphs. We prov...

We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ ℝ(d) and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y) = e(-y) for y ∈ ℝ; in this case, the resulting class of densities [Formula: see text]is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails tha...