We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees. Specifically, by leveraging a sophisticated characterization...

This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call ‘convex functors’; they include what are usually called probability distribution functors. These relationships take the form of three adjunctions. Two of these three are ‘dual’ adjunctions ...

Hermite-Hadamard inequality is one of the fundamental applications of convex functions in Theory of Inequality. In this paper, Hermite-Hadamard inequalities for $mathbb{B}$-convex and $mathbb{B}^{-1}$-convex functions are proven.

The scale and rotation invariance properties of a recently proposed algorithm, using the fuzzy evidence accumulation principle, for finding lines (ridges) of non-parametric shapes is analysed. The proposed modifications consist in scaling the accumulated value with the inverse of the line width and further fuzzifying the accumulation process – along the line width. Good invariance properties re...

We present an extension of convex-hull non-negative matrix factorization (CH-NMF) which was recently proposed as a large scale variant of convex non-negative matrix factorization or Archetypal Analysis. CH-NMF factorizes a non-negative data matrix V into two nonnegative matrix factors V ≈ WH such that the columns of W are convex combinations of certain data points so that they are readily inter...

in this paper we introduce a sequential block iterative method and its simultaneous version with op-timal combination of weights (instead of convex combination) for solving convex feasibility problems.when the intersection of the given family of convex sets is nonempty, it is shown that any sequencegenerated by the given algorithms converges to a feasible point. additionally for linear feasibil...

In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension 2n + 2 which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on ∂W , which we call convex open book, induced by a compact convex Lefschetz fibration on W carries the contact structure induced by the conv...

This paper describes the software DEpthLAUNAY. The main goal of the application is to compute Delaunay depth layers and levels of a planar point set [ACH]. Some other geometric structures can be computed as well (convex hull, convex layers and levels, Voronoi diagram and Voronoi levels, Delaunay triangulation, Delaunay empty circles, etc.) The application has been developed using CGAL [CGAL].

The aim of this paper is to characterize morphological convex geometries (resp., antimatroids). We define these two structures by using closure operators, and kernel operators. We show that these convex geometries are equivalent to poset geometries. 2000 Mathematics Subject Classification. 37F20, 06A07.