Algebraic signatures of convex and non-convex codes

Non-Convex Optimization via Real Algebraic Geometry

The high level purpose of this paper is to describe some recent advances in the field of Mathematics called Real Algebraic Geometry, and discuss some of its applications to complexity theory, and non-convex optimization. In particular, one of the questions underlying the entire development, is the crucial question: What makes an optimization problem difficult or easy? Along the way, we try to p...

Convex Non-convex Image Segmentation

A convex non-convex variational model is proposed for multiphase image segmen-tation. We consider a specially designed non-convex regularization term which adapts spatially tothe image structures for better controlling of the segmentation and easy handling of the intensityinhomogeneities. The nonlinear optimization problem is efficiently solved by an Alternating Direc-tions Meth...

Algebraic boundaries of convex semi-algebraic sets

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a com...

Non convex technologies and economic efficiency measure with imprecise data

Convex NMF on Non-Convex Massiv Data

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