Geometrical inverse preconditioning for symmetric positive definite matrices

We focus on inverse preconditioners based on minimizing $F(X) = 1-\cos(XA,I)$, where $XA$ is the preconditioned matrix and $A$ is symmetric and positive definite. We present and analyze gradient-type methods to minimize $F(X)$ on a suitable compact set. For that we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of $F(X)$ on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included

Cone-theoretic generalization of total positivity

This paper is devoted to the generalization of the theory of total positivity. We say that a linear operator A in R^n is generalized totally positive (GTP), if its jth exterior power preserves a proper cone K_j in the corresponding space for every j = 1, ..., n. We also define generalized strictly totally positive (GSTP) operators. We prove that the spectrum of a GSTP operator is positive and simple, moreover, its eigenvectors are localized in special sets. The existence of invariant cones of finite ranks is shown under some additional conditions. Some new insights and alternative proofs of the well-known results of Gantmacher and Krein describing the properties of TP and STP matrices are presented

On pairs of vectors achieving the maximal angle of a convex cone

International audienc

Towards parameter-less 3D mesh segmentation

This thesis focuses on the 3D mesh segmentation process. The research demonstrated how the process can be done in a parameterless approach which allows full automation with accurate results. Applications of this research include, but not limited to, 3D search engines, 3D character animation, robotics environment recognition, and augmented reality