Conditioning in Tropical Probability Theory.

We define a natural operation of conditioning of tropical diagrams of probability spaces and show that it is Lipschitz continuous with respect to the asymptotic entropy distance.

Arrow Contraction and Expansion in Tropical Diagrams.

Arrow contraction applied to a tropical diagram of probability spaces is a modification of the diagram, replacing one of the morphisms with an isomorphism while preserving other parts of the diagram. It is related to the rate regions introduced by Ahlswede and Körner. In a companion article, we use arrow contraction to derive information about the shape of the entropic cone.

Optimal Paths for Variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis.

We present a PDE-based approach for finding optimal paths for the Reeds-Shepp car. In our model we minimize a (data-driven) functional involving both curvature and length penalization, with several generalizations. Our approach encompasses the two- and three-dimensional variants of this model, state-dependent costs, and moreover, the possibility of removing the reverse gear of the vehicle.

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D.

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups SE(2) and SE(3). In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate SE(), and which are obtained via the exponential and logarithmic map on SE(). In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on SE().

From typical sequences to typical genotypes.

We demonstrate an application of a core notion of information theory, typical sequences and their related properties, to analysis of population genetic data. Based on the asymptotic equipartition property (AEP) for nonstationary discrete-time sources producing independent symbols, we introduce the concepts of typical genotypes and population entropy and cross entropy rate. We analyze three perspectives on typical genotypes: a set perspective on the interplay of typical sets of genotypes from two populations, a geometric perspective on their structure in high dimensional space, and a statistical learning perspective on the prospects of constructing typical-set based classifiers.