Warped Riemannian metrics for location-scale models

The present paper shows that warped Riemannian metrics, a class of Riemannian metrics which play a prominent role in Riemannian geometry, are also of fundamental importance in information geometry. Precisely, the paper features a new theorem, which states that the Rao-Fisher information metric of any location-scale model, defined on a Riemannian manifold, is a warped Riemannian metric, whenever this model is invariant under the action of some Lie group. This theorem is a valuable tool in finding the expression of the Rao-Fisher information metric of location-scale models defined on high-dimensional Riemannian manifolds. Indeed, a warped Riemannian metric is fully determined by only two functions of a single variable, irrespective of the dimension of the underlying Riemannian manifold. Starting from this theorem, several original contributions are made. The expression of the Rao-Fisher information metric of the Riemannian Gaussian model is provided, for the first time in the literature. A generalised definition of the Mahalanobis distance is introduced, which is applicable to any location-scale model defined on a Riemannian manifold. The solution of the geodesic equation is obtained, for any Rao-Fisher information metric defined in terms of warped Riemannian metrics. Finally, using a mixture of analytical and numerical computations, it is shown that the parameter space of the von Mises-Fisher model of $n$-dimensional directional data, when equipped with its Rao-Fisher information metric, becomes a Hadamard manifold, a simply-connected complete Riemannian manifold of negative sectional curvature, for $n = 2,\ldots,8$. Hopefully, in upcoming work, this will be proved for any value of $n$.Comment: first version, before submissio

A Critical Review of Multi-hole Drilling Path Optimization

Hole drilling is one of the major basic operations in part manufacturing. It follows without surprise then that the optimization of this process is of great importance when trying to minimize the total financial and environmental cost of part manufacturing. In multi-hole drilling, 70 % of the total process time is spent in tool movement and tool switching. Therefore, toolpath optimization in particular has attracted significant attention in cost minimization. This paper critically reviews research publications on drilling path optimization. In particular, this review focuses on three aspects; problem modeling, objective functions, and optimization algorithms. We conclude that most papers being published on hole drilling are simply basic Traveling Salesman Problems (TSP) for which extremely powerful heuristics exist and for which source code is readily available. Therefore, it is remarkable that many researchers continue developing “novel” metaheuristics for hole drilling without properly situating those approaches in the larger TSP literature. Consequently, more challenging hole drilling applications that are modeled by the Precedence Constrained TSP or hole drilling with sequence dependent drilling times do not much research focus. Sadly, these many low quality hole drilling research publications drown out the occasional high quality papers that describe specific problematic problem constraints or objective functions. It is our hope through this review paper that researchers’ efforts can be refocused on these problem aspects in order to minimize production costs in the general sense