Contractions in non-Euclidean spaces

Spatial Analysis in curved spaces with Non-Euclidean Geometry

The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type an...

Windows into non-Euclidean spaces

Cellular Automata in Non-Euclidean Spaces

Classical results on the surjectivity and injectivity of parallel maps are shown to be extendible to the cases with non-Euclidean cell spaces of particular types. Also shown are obstructions to extendibility, which may shed light on the nature of classical results such as the Garden-of-Eden theorem. Key–Words: Cellular automata, Non Euclidean cell spaces, the Garden-of-Eden theorem

Multi-Manifold Modeling in Non-Euclidean spaces

This paper advocates a novel framework for segmenting a dataset on a Riemannian manifold M into clusters lying around lowdimensional submanifolds of M . Important examples of M , for which the proposed algorithm is computationally efficient, include the sphere, the set of positive definite matrices, and the Grassmannian. The proposed algorithm constructs a data-affinity matrix by thoroughly exp...

Neural networks in non-Euclidean metric spaces

Multilayer Perceptrons (MLPs) use scalar products to compute weighted activation of neurons providing decision borders using combinations of soft hyperplanes. The weighted fun-in activation function corresponds to Euclidean distance functions used to compute similarities between input and weight vector. Replacing the fan-in activation function by non-Euclidean distance function offers a natural...